Hybrid and Multifield Inflation
by
Evangelos I. Sfakianakis
Submitted to the Department of Physics
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
c Massachusetts Institute of Technology 2014. All rights reserved.
Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Department of Physics
May 22, 2014
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alan H. Guth
Victor F. Weisskopf Professor of Physics
Thesis Supervisor
Certified by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
David I. Kaiser
Germeshausen Professor of the History of Science
and Senior Lecturer, Department of Physics
Thesis Supervisor
Accepted by . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Krishna Rajagopal
Associate Department Head for Education
2
Hybrid and Multifield Inflation
by
Evangelos I. Sfakianakis
Submitted to the Department of Physics
on May 22, 2014, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Abstract
In this thesis I study the generation of density perturbations in two classes of inflationary models: hybrid inflation and multifield inflation with nonminimal coupling to
gravity.
• In the case of hybrid inflation, we developed a new method of treating these
perturbations that does not rely on a classical trajectory for the fields. A
characteristic of the spectrum is the appearance of a spike at small length scales,
which could conceivably seed the formation of black holes that can evolve to
become the supermassive black holes found at the centers of galaxies. Apart
from numerically calculating the resulting spectrum, we derived an expansion
in the number of waterfall fields, which makes the calculation easier and more
intuitive.
• In the case of multifield inflation, we studied models where the scalar fields
are coupled non-minimally to gravity. We developed a covariant formalism and
examined the prediction for non-Gaussianities in these models, arguing that
they are absent except in the case of fine-tuned initial conditions. We have also
applied our formalism to Higgs inflation and found that multifield effects are
too small to be observable. We compared these models to the early data of the
Planck satellite mission, finding excellent agreement for the spectral index and
tensor to scalar ratio and promising agreement for the existence of isocurvature
modes.
Thesis Supervisor: Alan H. Guth
Title: Victor F. Weisskopf Professor of Physics
Thesis Supervisor: David I. Kaiser
Title: Germeshausen Professor of the History of Science
and Senior Lecturer, Department of Physics
3
4
Acknowledgments
I am grateful to my supervisors, Prof. Alan Guth and Prof. David Kaiser, for their
support and mentoring. I thank my thesis readers, Prof. Hong Liu and Prof. Max
Tegmark for their advice.
I also thank my collaborators, Prof. Ruben Rosales,
Prof. Edward Farhi, Prof. Iain Stewart, Dr. Mark Hertzberg, Dr. Mustafa Amin,
Ross Greenwood, Illan Halpern, Matthew Joss, Edward Mazenc and Katelin Schutz
for their contribution to this thesis and beyond.
I would like to thank my family and friends for their continued support and
Angeliki for being there for me during the last two years.
I am indebted to my fellow students, the staff of the CTP, Joyce Berggren, Scott
Morley and Charles Suggs and the staff of the Physics Department for creating a
vibrant environment. I thank the undergraduate students who worked with us, since
they reminded me why I wanted to be a physicist, at a time when I needed it the
most. I would not be here without the teachers and professors, in Greece and the US,
who inspired me and I hope I have met their expectations.
Last but not least I would like to thank George J. Elbaum and Mimi Jensen for
their generosity that gave me the opportunity to pursue a PhD at MIT.
This thesis is dedicated to the memory of my beloved grandmother.
5
6
Contents
1 Introduction
1.1
1.2
1.3
25
Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1.1.1
Simple Single field inflation . . . . . . . . . . . . . . . . . . .
27
1.1.2
Inflationary Perturbations . . . . . . . . . . . . . . . . . . . .
28
Beyond the single field . . . . . . . . . . . . . . . . . . . . . . . . . .
30
1.2.1
Hybrid Inflation . . . . . . . . . . . . . . . . . . . . . . . . . .
31
1.2.2
Non-minimal coupling . . . . . . . . . . . . . . . . . . . . . .
31
Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . .
33
2 Density Perturbations in Hybrid Inflation Using a Free Field Theory
Time-Delay Approach
37
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
2.2
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.2.1
Field set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
2.2.2
Fast Transition . . . . . . . . . . . . . . . . . . . . . . . . . .
44
2.2.3
Mode expansion . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.2.4
Solution of the mode function . . . . . . . . . . . . . . . . . .
46
Supernatural Inflation models . . . . . . . . . . . . . . . . . . . . . .
52
2.3.1
55
2.3
2.4
2.5
End of Inflation . . . . . . . . . . . . . . . . . . . . . . . . . .
Perturbation theory basics
. . . . . . . . . . . . . . . . . . . . . . .
56
2.4.1
Time delay formalism . . . . . . . . . . . . . . . . . . . . . . .
56
2.4.2
Randall-Soljacic-Guth approximation . . . . . . . . . . . . . .
57
Calculation of the Time Delay Power Spectrum . . . . . . . . . . . .
59
7
2.6
Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . .
64
2.6.1
Model-Independent Parameter Sweep . . . . . . . . . . . . . .
66
2.6.2
Supernatural Inflation . . . . . . . . . . . . . . . . . . . . . .
71
2.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
2.8
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
2.9
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
2.9.1
Zero mode at early times . . . . . . . . . . . . . . . . . . . . .
78
2.9.2
Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . .
80
3 Hybrid Inflation with N Waterfall Fields: Density Perturbations
and Constraints
87
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3.2
N Field Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
90
3.2.1
Approximations . . . . . . . . . . . . . . . . . . . . . . . . . .
92
3.2.2
Mode Functions . . . . . . . . . . . . . . . . . . . . . . . . . .
94
3.3
The Time-Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
3.4
Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
3.4.1
Two-Point Function
99
3.4.2
Three-Point Function . . . . . . . . . . . . . . . . . . . . . . . 102
3.4.3
Momentum Space . . . . . . . . . . . . . . . . . . . . . . . . . 105
3.5
. . . . . . . . . . . . . . . . . . . . . . .
Constraints on Hybrid Models . . . . . . . . . . . . . . . . . . . . . . 107
3.5.1
Topological Defects . . . . . . . . . . . . . . . . . . . . . . . . 107
3.5.2
Inflationary Perturbations . . . . . . . . . . . . . . . . . . . . 108
3.5.3
Implications for Scale of Spike . . . . . . . . . . . . . . . . . . 109
3.5.4
Eternal Inflation . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.6
Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 112
3.7
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.8
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.8.1
Series Expansion to Higher Orders . . . . . . . . . . . . . . . 114
3.8.2
Two-Point Function for Even Number of Fields . . . . . . . . 115
8
3.8.3
Alternative Derivation of Time-Delay Spectra . . . . . . . . . 116
4 Primordial Bispectrum from Multifield Inflation with Nonminimal
Couplings
123
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.2
Evolution in the Einstein Frame . . . . . . . . . . . . . . . . . . . . . 126
4.3
Covariant Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
4.4
Adiabatic and Entropy Perturbations . . . . . . . . . . . . . . . . . . 144
4.5
Primordial Bispectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
4.7
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.8
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.8.1
Field-Space Metric and Related Quantities . . . . . . . . . . . 164
5 Multifield Dynamics of Higgs Inflation
175
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.2
Multifield Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.3
Application to Higgs Inflation . . . . . . . . . . . . . . . . . . . . . . 182
5.4
Turn Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
5.5
Implications for the Primordial Spectrum . . . . . . . . . . . . . . . . 197
5.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
5.7
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
5.8
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5.8.1
Angular Evolution of the Field
. . . . . . . . . . . . . . . . . 203
6 Multifield Inflation after Planck:
The Case for Nonminimal Couplings
211
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
6.2
Multifield Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
6.3
The Single-Field Attractor . . . . . . . . . . . . . . . . . . . . . . . . 217
6.4
Observables and the Attractor Solution . . . . . . . . . . . . . . . . . 219
9
6.5
Isocurvature Perturbations . . . . . . . . . . . . . . . . . . . . . . . . 221
6.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
6.7
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
7 Multifield Inflation after Planck : Isocurvature Modes from Nonminimal Couplings
229
7.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
7.2
Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.3
7.4
7.2.1
Einstein-Frame Potential . . . . . . . . . . . . . . . . . . . . . 233
7.2.2
Coupling Constants . . . . . . . . . . . . . . . . . . . . . . . . 236
7.2.3
Dynamics and Transfer Functions . . . . . . . . . . . . . . . . 238
Trajectories of Interest . . . . . . . . . . . . . . . . . . . . . . . . . . 243
7.3.1
Geometry of the Potential . . . . . . . . . . . . . . . . . . . . 243
7.3.2
Linearized Dynamics . . . . . . . . . . . . . . . . . . . . . . . 247
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
7.4.1
Local curvature of the potential . . . . . . . . . . . . . . . . . 252
7.4.2
Global structure of the potential . . . . . . . . . . . . . . . . . 254
7.4.3
Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 256
7.4.4
CMB observables . . . . . . . . . . . . . . . . . . . . . . . . . 257
7.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
7.6
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
7.7
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
7.7.1
Approximated Dynamical Quantities . . . . . . . . . . . . . . 265
7.7.2
Covariant formalism and potential topography . . . . . . . . . 267
10
List of Figures
1-1 A pictorial form of the hybrid potential. . . . . . . . . . . . . . . . .
32
2-1 Mode functions for different comoving wavenumbers as a function of
time in efolds. The model parameters are µψ =
1
10
and µφ = 10. We
can see the modes following our analytic approximation for the growth
rate. Our analysis gives Ndev (1/256) ≈ −7.9 and Ntr (256) ≈ 4.76,
which are very close to the values that can be read off the graph.
. .
51
2-2 Mode functions for different comoving wavenumbers as a function of
time in efolds. The model parameters are µψ =
1
2
and µφ = 1. The
horizontal line corresponds to the asymptotic value of the growth factor λ. We can see how the mode functions reach their asymptotic
behavior after 10 efolds. Our analysis gives Ndev (1/256) ≈ −7.84 and
Ntr (256) ≈ 5.4, which are very close to the values that can be read off
the graph. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
2-3 Parameter space for the two supernatural inflation models. The bottom right corner shows the parameter for model 2, while the other
three show parameters for model 1, for different ranges of the mass
scale M 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
2-4 The time delay field calculated using the RSG formalism. The end time
was taken to be 15 e-folds after the waterfall transition and µψ =
1
20
for all curves, while we varied µφ . . . . . . . . . . . . . . . . . . . . .
11
59
2-5 Comparison between the RSG and FFT methods. The end time was
taken to be 15 e-folds after the waterfall transition and µφ = 20 and
µψ = 1/20. We can see that the spectrum of the time delay field
calculated in the free field theory agrees very well with the rescaled
version of the RSG approximation A δτRSG (Bk).
. . . . . . . . . . .
65
2-6 Parameter sweep for constant timer field mass µψ = 1/20 and constant
end field value φend = 1014 . Data points are plotted along with a least
square power law fit. The same trend is evident in all curves. The time
delay spectrum grows in amplitude and width and is shifted towards
larger momentum values as the mass product decreases. Also inflation
takes longer to end for low mass product. . . . . . . . . . . . . . . . .
67
2-7 Time delay spectra for different values of the mass product, keeping
the timer field mass fixed at µψ =
1
20
. . . . . . . . . . . . . . . . . .
68
2-8 Perturbation spectrum for varying field value at tend for constant masses.
The time delay curves are identical in shape and differ only in amplitude. This variation is entirely due to the different value of the time
dependent growth factor λ, which differs for each case because inflation
simply takes longer to end for larger end field values. . . . . . . . . .
69
2-9 Fixing the mass product at 2 and varying the mass ratio. There is
significant variation only for low mass ratio, when the light timer field
approximation loses its validity. Furthermore the curves of maximum
time delay amplitude and 1/λ follow each other exactly up to our
numerical accuracy. Finally by rescaling the spectra by the growth
factor λ they become identical for all values of the mass ratio. . . . .
72
2-10 Fixing the mass ratio at 900 (open circles) or the timer mass at µψ =
1/20 (+’s). The time delay spectra for different mass products show no
common shape characteristics and remain different even when rescaled
by λ. Furthermore the end time and maximum perturbation amplitude
curves are identical for constant mass ratio and constant timer field
mass, proving that indeed the mass product is the dominant parameter. 73
12
2-11 First Supernatural inflation model with M 0 at the Planck scale. The
spectra corresponding to the maximum and minimum mass product
are shown. We observe good agreement with the results of the model
independent parameter sweep of the previous section, because the timer
field mass is much smaller than the Hubble scale. . . . . . . . . . . .
74
2-12 First Supernatural inflation model with M 0 at the GUT scale. The
spectra corresponding to the maximum and minimum mass product
are shown. There is again good agreement with the results of the
previous section. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
2-13 First Supernatural inflation model with M 0 at the intermediate scale.
Five representative pairs of masses were chosen and the corresponding
time delay curves are shown. This model can give maximum time delay
of more than 0.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
2-14 Second Supernatural inflation model. Three representative pairs of
masses were chosen and the corresponding time delay curves are shown. 76
2-15 Mode functions for µφ = 22 and m̃uψ = 1/18. The left column is
calculated for k̃ = 1/256 and the right for k̃ = 256
. . . . . . . . . .
82
3-1 An illustration of the evolution of the effective potential for the waterfall field φ as the timer field ψ evolves from “high” values at early
times, to ψ = ψc , and finally to “low” values at late times. In the
process, the effective mass-squared of φ evolves from positive, to zero,
to negative (tachyonic). . . . . . . . . . . . . . . . . . . . . . . . . . .
92
3-2 A plot of the (re-scaled) field’s power spectrum Pφ̃ as a function of
wavenumber k (in units of H) for different masses: blue is m0 = 2
and mψ = 1/2, red is m0 = 4 and mψ = 1/2, green is m0 = 2 and
mψ = 1/4, orange is m0 = 4 and mψ = 1/4. . . . . . . . . . . . . . . .
13
95
3-3 A plot of the field’s correlation ∆ as a function of x (in units of H −1 )
for different masses: blue is m0 = 2 and mψ = 1/2, red is m0 = 4
and mψ = 1/2, green is m0 = 2 and mψ = 1/4, orange is m0 = 4 and
mψ = 1/4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
3-4 Top: a plot of the (re-scaled) two-point function of the time-delay
N (2λ)2 hδt(x)δt(0)i as a function of ∆ ∈ [0, 1] as we vary N : dotdashed is N = 1, solid is N = 2, dotted is N = 6, and dashed is
N → ∞. Bottom: a plot of the (re-scaled) two-point function of the
time-delay N hδt(x)δt(0)i as a function of x for different masses: blue
is m0 = 2 and mψ = 1/2, red is m0 = 4 and mψ = 1/2, green is m0 = 2
and mψ = 1/4, orange is m0 = 4 and mψ = 1/4, with solid for N = 2
and dashed for N → ∞. . . . . . . . . . . . . . . . . . . . . . . . . . 103
3-5 The dimensionless power spectrum N Pδt (k) at large N as a function
of k (in units of H) for different choices of masses: blue is m0 = 2
and mψ = 1/2, red is m0 = 4 and mψ = 1/2, green is m0 = 2 and
mψ = 1/4, orange is m0 = 4 and mψ = 1/4. . . . . . . . . . . . . . . . 106
3-6 The dimensionless bispectrum
√
N FN L as a function of k (in units of
H) at large N for m0 = 4 and mψ = 1/2. . . . . . . . . . . . . . . . . 107
4-1 The Einstein-frame effective potential, Eq. (4.18), for a two-field model.
The potential shown here corresponds to the couplings ξχ /ξφ = 0.8, λχ /λφ =
2. . . . . . . . . . . . . . . .
0.3, g/λφ = 0.1, and m2φ = m2χ = 10−2 λφ Mpl
14
132
4-2 Parametric plot of the fields’ evolution superimposed on the Einstein-frame
potential. Trajectories for the fields φ and χ that begin near the top of
a ridge will diverge. In this case, the couplings of the potential are ξφ =
10, ξχ = 10.02, λχ /λφ = 0.5, g/λφ = 1, and mφ = mχ = 0. (We use
p
a dimensionless time variable, τ ≡ λφ Mpl t, so that the Jordan-frame
couplings are measured in units of λφ .) The trajectories shown here each
have the initial condition φ(τ0 ) = 3.1 (in units of Mpl ) and different values
of χ(τ0 ): χ(τ0 ) = 1.1 × 10−2 (“trajectory 1,” yellow dotted line); χ(τ0 ) =
1.1×10−3 (“trajectory 2,” red solid line); and χ(τ0 ) = 1.1×10−4 (“trajectory
3,” black dashed line). . . . . . . . . . . . . . . . . . . . . . . . . . . .
133
4-3 The evolution of the Hubble parameter (black dashed line) and the background fields, φ(τ ) (red solid line) and χ(τ ) (blue dotted line), for trajectory
2 of Fig. 2. (We use the same units as in Fig. 2, and have plotted 100H
so its scale is commensurate with the magnitude of the fields.) For these
couplings and initial conditions the fields fall off the ridge in the potential
at τ = 2373 or N = 66.6 efolds, after which the system inflates for another
4.9 efolds until τend = 2676, yielding Ntotal = 71.5 efolds. . . . . . . . . .
134
4-4 Models with nonzero masses include additional features in the Einsteinframe potential which can also cause neighboring field trajectories to diverge.
In this case, we superimpose the evolution of the fields φ and χ on the
Einstein-frame potential. The parameters shown here are identical to those
2 and m2 = 0.0025 λ M 2 rather than
in Fig. 4-2 but with m2φ = 0.075 λφ Mpl
φ
χ
pl
0. The initial conditions match those of trajectory 3 of Fig. 4-2: φ(τ0 ) = 3.1
and χ(τ0 ) = 1.1 × 10−4 in units of Mpl .
. . . . . . . . . . . . . . . . . . 135
4-5 Parametric plot of the evolution of the fields φ and χ superimposed on the
Ricci curvature scalar for the field-space manifold, R, in the absence of a
Jordan-frame potential. The fields’ geodesic motion is nontrivial because of
the nonvanishing curvature. Shown here is the case ξφ = 10, ξχ = 10.02,
φ(τ0 ) = 0.75, χ(τ0 ) = 0.01, φ0 (τ0 ) = −0.01, and χ0 (τ0 ) = 0.005.
15
. . . . . . 136
4-6 The slow-roll parameters (blue dashed line) and |ησσ | (solid red line) versus
N∗ for trajectory 2 of Fig. 2, where N∗ is the number of efolds before the
end of inflation. Note that |ησσ | temporarily grows significantly larger than
1 after the fields fall off the ridge in the potential at around N∗ ∼ 5. . . .
142
4-7 The turn-rate, ω = |ω I |, for the three trajectories of Fig. 4-2: trajectory 1
(orange dotted line); trajectory 2 (red solid line); and trajectory 3 (black
dashed line). The rapid oscillations in ω correspond to oscillations of φ
in the lower false vacuum of the χ valley. For trajectory 1, ω peaks at
N∗ = 34.5 efolds before the end of inflation; for trajectory 2, ω peaks at
N∗ = 4.9 efolds before the end of inflation; and for trajectory 3, ω remains
much smaller than 1 for the duration of inflation. . . . . . . . . . . . . .
143
4-8 The effective mass-squared of the entropy perturbations relative to the Hubble scale, (µs /H)2 , for the trajectories shown in Fig. 4-2: trajectory 1
(orange dotted line); trajectory 2 (red solid line); and trajectory 3 (black
dashed line). For all three trajectories, µ2s < 0 while the fields remain near
the top of the ridge, since µ2s is related to the curvature of the potential in
the direction orthogonal to the background fields’ evolution. The effective
mass grows much larger than H as soon as the fields roll off the ridge of the
potential.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
4-9 The transfer function TRS for the three trajectories of Fig. 4-2: trajectory
1 (orange dotted line); trajectory 2 (red solid line); and trajectory 3 (black
dashed line). Trajectories 2 and 3, which begin nearer the top of the ridge
in the potential than trajectory 1, evolve as essentially single-field models
during early times, before the fields roll off the ridge. . . . . . . . . . . .
153
4-10 The spectral index, ns , versus N∗ for the three trajectories of Fig. 4-2:
trajectory 1 (orange dotted line); trajectory 2 (red solid line); and trajectory
3 (black dashed line). The spectral indices for trajectories 2 and 3 coincide
and show no tilt from the value ns (N60 ) = 0.967. . . . . . . . . . . . . .
16
154
4-11 The non-Gaussianity parameter, |fN L |, for the three trajectories of Fig.
4-2: trajectory 1 (orange dotted line); trajectory 2 (solid red line); and
trajectory 3 (black dashed line). Changing the fields’ initial conditions by a
small amount leads to dramatic changes in the magnitude of the primordial
bispectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
162
5-1 The potential for Higgs inflation in the Einstein frame, V (φ, χ). Note the
flattening of the potential for large field values, which is quite distinct from
the behavior of the Jordan-frame potential, Ṽ (φ, χ) in Eq. (5.27). . . . . .
184
5-2 The evolution of H(t) (black dashed line) and the fields φ(t) (red solid line)
and χ(t) (blue dotted line). The fields are measured in units of Mpl and we
√
use the dimensionless time variable τ = λ Mpl t. We have plotted 103 H so
that its scale is commensurate with the magnitude of the fields. The Hubble
parameter begins large, H(0) = 8.1 × 10−4 , but quickly falls by a factor of
30 as it settles to its slow-roll value of H = 2.8×10−5 . Inflation proceeds for
∆τ = 2.5 × 106 to yield N = 70.7 efolds of inflation. The solutions shown
here correspond to ξ = 104 , φ(0) = 0.1, χ(0) = 0, φ̇(0) = −2 × 10−6 , and
χ̇(0) = 2 × 10−2 . For the same value of φ(0), Eq. (5.40) corresponds to
φ̇(0) = −2 × 10−8 for the single-field case.
. . . . . . . . . . . . . . . . 188
5-3 Contour plots showing the number of efolds of inflation as one varies the
fields’ initial conditions, keeping ξ = 104 fixed. In each panel, the vertical
axis is χ̇(0) and the horizontal axis is φ̇(0). The panels correspond to φ(0) =
10−1 Mpl (top left), 10−2 Mpl (top right), 5 × 10−3 Mpl (bottom left) and
√
10−4 Mpl (bottom right), and we again use dimensionless time τ = λ Mpl t.
In each panel, the line for N = 70 efolds is shown in bold. Note how large
these initial velocities are compared to the single-field expectation of Eq.
(5.40). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
189
5-4 Evolution of the turn rate. The left picture shows the evolution with initial
conditions as in Fig. 5-2. The right figure has initial conditions φ(0) = 0.1,
√
χ(0) = φ̇(0) = 0, and χ̇(0) = 2 × 10−5 in units of Mpl and τ = λ Mpl t.
In both cases we set ξ = 104 . Recall from Fig. 5-2 that inflation lasts until
τend ∼ O(106 ) for these initial conditions; hence we find that ω damps out
within a few efolds after the start of inflation. . . . . . . . . . . . . . . .
190
5-5 Contour plots showing the number of efolds at which the maximum of the
turn rate occurs, as one varies the fields’ initial conditions. In each panel, the
vertical axis is χ̇(0) and the horizontal axis is φ̇(0). The panels correspond
to φ(0) = 10−1 Mpl (top left), 10−2 Mpl (top right), 5×10−3 Mpl (bottom left)
and 10−4 Mpl (bottom right). We set ξ = 104 and use the dimensionless
√
time-variable τ = λ Mpl t. The thick black curve is the contour line of
initial conditions that yield N = 70 efolds. . . . . . . . . . . . . . . . . .
194
5-6 The turn rate as a function of time for different values of the initial angular
velocity. The parameters used are ξ = 104 , φ(0) = 0.1Mpl , φ̇(0) = χ(0) = 0,
√
0.01
100
and √
≤ γ̇(0) ≤ √
, in terms of dimensionless time, τ = λ Mpl t. In
2ξ
2ξ
these units and for φ(0) = 0.1Mpl , inflation lasts until τend ∼ O(106 ).
. . 195
5-7 Number of efolds until the maximum value of the turn rate is reached, as
a function of χ̇(0). On the left we plot N (ωmax ) for initial conditions that
yield Ntot = 70 efolds; on the right we plot the same quantity for various
values of φ̇(0). The logarithmic fit is an excellent match to our analytic
result, Eq. (5.61).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
18
5-8 Dynamics of our two-field model with ξ = 102 , φ(0) = 1Mpl , and χ(0) = 0.
Clockwise from top left: (1) Contour plot showing the number of efolds
as one varies the fields’ initial conditions. The thick curve corresponds to
70 efolds. (2) Contour plot showing the number of efolds at which the
maximum of the turn rate occurs, as one varies the fields’ initial conditions.
The thick curve corresponds to Ntot = 70 efolds. (3) Number of efolds
until the maximum value of the turn rate is reached for initial conditions
giving Ntot = 70 efolds, along with a logarithmic fit. (4) The turn rate as
a function of time for different values of the initial angular velocity, with
√
0.01
100
φ̇(0) = 0 and √
≤ γ̇(0) ≤ √
, in units of τ = λ Mpl t. . . . . . . . . .
2ξ
2ξ
198
6-1 Potential in the Einstein frame, V (φI ). Left: λχ = 0.75 λφ , g = λφ , ξχ =
1.2 ξφ . Right: λχ = g = λφ , ξφ = ξχ . In both cases, ξI 1 and 0 < λI , g < 1. 215
6-2 Left: Field trajectories for different couplings and initial conditions (here
for φ̇0 , χ̇0 = 0). Open circles indicate fields’ initial values. The parameters
{λχ , g, ξχ , θ0 } are given by: {0.75λφ , λφ , 1.2ξφ , π/4} (red), {λφ , λφ , 0.8ξφ , π/4}
(blue), {λφ , 0.75 λφ , 0.8ξφ , π/6} (green), {λφ , 0.75 λφ , 0.8ξφ , π/3} (black).
Right: Numerical vs. analytic evaluation of the slow-roll parameters, (numerical = blue, analytic = red) and ησσ (numerical = black, analytic =
green), for λφ = 0.01, λχ = 0.75 λφ , g = λφ , ξφ = 103 , and ξχ = 1.2 ξφ , with
θ0 = π/4 and φ̇0 = χ̇0 = +10 |φ̇SR |.
. . . . . . . . . . . . . . . . . . . . 219
7-1 Potential in the Einstein frame, V (φI ) in Eq. (7.7). The parameters
shown here are λχ = 0.75 λφ , g = λφ , ξχ = 1.2 ξφ , with ξφ 1 and
λφ > 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
19
7-2 The spectral index ns (red), as given in Eq. (7.61), for different values
of κ, which characterizes the local curvature of the potential near the
top of a ridge. Also shown are the 1σ (thin, light blue) and 2σ (thick,
dark blue) bounds on ns from the Planck measurements. The couplings
shown here correspond to ξφ = ξχ = 103 , λφ = 10−2 , and λχ = g, fixed
for a given value of κ from Eq. (7.10). The fields’ initial conditions are
φ = 0.3, φ̇0 = 0, χ0 = 10−3 , χ̇0 = 0, in units of Mpl . . . . . . . . . . . 235
7-3 The fields’ trajectory (red) superimposed upon the effective potential
in the Einstein frame, V , with couplings ξφ = 1000, ξχ = 1000.015,
λφ = λχ = g = 0.01, and initial conditions φ0 = 0.35, χ0 = 8.1 × 10−4 ,
φ̇0 = χ̇0 = 0, in units of Mpl .
. . . . . . . . . . . . . . . . . . . . . . 244
7-4 The mass of the isocurvature modes, µ2s /H 2 (blue, solid), and the turn
rate, (ω/H)×103 (red, dotted), versus e-folds from the end of inflation,
N∗ , for the trajectory shown in Fig. 7-3. Note that while the fields
ride along the ridge, the isocurvature modes are tachyonic, µ2s < 0,
leading to an amplification of isocurvature perturbations. The mass µ2s
becomes large and positive once the fields roll off the ridge, suppressing
further growth of isocurvature modes.
. . . . . . . . . . . . . . . . . 245
7-5 The asymptotic value r → ∞ for three potentials with Λχ = −0.001
(blue dashed), Λχ = 0 (red solid), and Λχ = 0.001 (yellow dotted), as a
function of the angle θ = arctan(χ/φ). For all three cases, Λφ = 0.0015,
ξφ = ξχ = 1000, and λφ = 0.01. . . . . . . . . . . . . . . . . . . . . . . 246
7-6 The evolution of TSS (top) and TRS (bottom) using the exact and
approximated expressions, for κ ≡ 4Λφ /λφ = 0.06, 4Λχ /λχ = −0.06
and ε = −1.5 × 10−5 , with φ0 = 0.35 Mpl , χ0 = 8.1 × 10−4 Mpl , and
φ̇0 = χ̇0 = 0. We take Nhc = 60 and plot TSS and TRS against N∗ , the
number of e-folds before the end of inflation. The approximation works
particularly well at early times and matches the qualitative behavior
of the exact numerical solution at late times. . . . . . . . . . . . . . . 251
20
7-7 The isocurvature fraction for different values of χ0 (in units of Mpl )
as a function of the curvature of the ridge, κ. All of the trajectories
began at φ0 = 0.3 Mpl , which yields Ntot = 65.7. For these potentials,
ξφ = 1000, λφ = 0.01, ε = 0, and Λχ = 0. The trajectories that begin
closest to the top of the ridge have the largest values of βiso , with some
regions of parameter space nearly saturating βiso = 1. . . . . . . . . . 253
7-8 Contributions of TRS and TSS to βiso . The parameters used are φ0 =
0.3 Mpl , χ0 = 10−3 Mpl , ξφ = 103 , λφ = 0.01, ε = 0 and Λχ = 0.
For small κ, βiso is dominated by TSS ; for larger κ, TRS becomes more
important and ultimately reduces βiso . . . . . . . . . . . . . . . . . . 254
7-9 The isocurvature fraction for different values of Λχ as a function of the
curvature of the ridge, κ. All of the trajectories began at φ0 = 0.3 Mpl
and χ0 = 10−4 Mpl , yielding Ntot = 65.7. For these potentials, ξφ =
1000, λφ = 0.01, and ε = 0. Potentials with Λχ < 0 yield the largest
βiso peaks, though in those cases βiso falls fastest in the large-κ limit
due to sensitive changes in curvature along the trajectory. Meanwhile,
potentials with positive Λχ suppress the maximum value of βiso once
κ & 0.1 and local curvature becomes important. . . . . . . . . . . . . 255
7-10 The isocurvature fraction for different values of ε as a function of the
curvature of the ridge, κ. All of the trajectories began at χ0 = 10−3 Mpl
and φ0 = 0.3 Mpl , with Ntot = 65.7. For these potentials, ξφ = 1000,
λφ = 0.01, and Λχ = 0. Here we see the competition between ε setting
the scale of the isocurvature mass and affecting the amount of turning
in field-space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256
21
7-11 Numerical simulations of βiso for various initial conditions (in units
of Mpl ). All trajectories shown here were for a potential with κ =
4Λφ /λφ = 0.116, 4Λχ /λχ = −160.12, and ε = −2.9 × 10−5 . Also shown
are our analytic predictions for contours of constant βiso , derived from
Eq. (7.58) and represented by dark, solid lines. From top right to
bottom left, the contours have βiso = 0.071, 0.307, 0.054, 0.183, and
0.355. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
7-12 Two trajectories from Fig. 7-11 that lie along the βiso = 0.183 line,
for φ0 = 0.3 Mpl and φ0 = 0.365 Mpl . The dots mark the fields’ initial
values. The two trajectories eventually become indistinguishable, and
hence produce identical values of βiso . . . . . . . . . . . . . . . . . . . 259
7-13 The spectral index ns for different values of the local curvature κ. The
parameters used are φ0 = 0.3Mpl , χ0 = 10−3 Mpl , ξφ = 1000, λφ = 0.01,
ε = 0 and Λχ = 0. Comparing this with Fig. 7-7 we see that the peak
in the βiso curve occurs within the Planck allowed region.
. . . . . . 261
7-14 The tensor-to-scalar ratio as a function of the local curvature parameter
κ. The parameters used are φ0 = 0.3 Mpl , χ0 = 10−3 Mpl , ξφ = 1000,
λφ = 0.01, ε = 0 and Λχ = 0. . . . . . . . . . . . . . . . . . . . . . . . 262
7-15 The correlation between r and ns could theoretically break the degeneracy between our models. The parameters used for this plot are
φ0 = 0.3 Mpl , χ0 = 10−3 Mpl , ξφ = 1000, λφ = 0.01, ε = 0 and Λχ = 0,
with 0 ≤ κ ≤ 0.1.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
22
List of Tables
2.1
5.1
Parameters of SUSY models and their counterparts in our quadratic
approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
Numerical results for measures of multifield dynamics for Higgs inflation
√
with ξ = 104 . We use dimensionless time τ = λMpl t. . . . . . . . . . .
201
23
24
Chapter 1
Introduction
1.1
Inflation
Inflation was introduced by Alan Guth in 1980 [1], and developed into a full working
model by Linde [2] and Albrecht and Steinhardt [3] in 1981 and 1982. We will
start the thesis by listing what can be considered the four main supporting pieces
of evidence for inflation. We will not include the absence of monopoles from our
universe, although this was one of the initial motivations.
1. Uniformity at large scales: If one examines the universe at large scales, one
can see that it is approximately homogenous. From the CMB to the distribution
of galaxies, the universe looks the same no matter “which way we look”. Based
on the standard big bang theory and the fact that information travels at the
speed of light at most, opposite parts of the universe were not in causal contact
in the past, hence they are not expected to be uniform to that degree (1 in
100, 000). Inflation solves this problem by starting with a very small uniform
patch and doubling it 100 times to create our observable universe.
2. Flatness of the universe: To explain the flatness problem of the old big bang
theory, we should start from an isotropic universe governed by the Friedman25
Robertson-Walker metric
2
2
ds = dt − a(t)
2
dr2
2
2
2
2
+ r (dθ + sin θdφ ) ,
1 − kr2
(1.1)
where a(t) is the scale factor and k defines the curvature of the universe. Let
us define the ratio
Ω=
mass density
.
critical mass density
(1.2)
The parameter |Ω − 1| defines the departure from flatness (Ω = 1 for a flat
universe). However |Ω − 1| is increasing after the big bang. In fact, in order for
the observed value of Ω = 1 + 0.0010 ± 0.0065 [4] to be possible, Ω had to be
exponentially close to unity at the big bang. Indeed inflation shows that
|Ωend − 1|
=
|Ωinit − 1|
ainit
aend
2
= e−2Ne . 10−50 ,
(1.3)
meaning that inflation can start with a patch of some curvature and the resulting
universe will be approximately flat to a high degree.
3. Perturbations: Although the universe is made uniform by inflation, we can
see non-uniformities all around us, from planets to solar systems, from galaxies
to galaxy clusters and of course in the CMB. This can also be explained quite
naturally by inflation. One has to take a step back and realize that quantum
mechanics demands that field values are fluctuating on very short scales. During inflation these fluctuations are stretched along with the rest of the universe,
so they reach cosmological dimensions. These fluctuations are observable in
the CMB, and they also provide the seeds of all structure around us, as they
are then enhanced by the unstable nature of gravitational systems. Inflationary predictions agree with the measured spectrum of these fluctuations to an
excellent degree [5].
4. Tensor modes: Inflation has the ability to stretch fluctuations of the gravitational field that exist at the quantum level to cosmological scales. The am26
plitude of the resulting primordial gravity waves is proportional to the energy
scale during inflation. Suppression of gravity waves would not rule out inflation, but rather point to models of “low-scale” inflation. However a detection of
primordial gravity waves (through B-modes in the CMB) has been considered
a smoking gun for inflation. Such a discovery was announced as the present
thesis was being written [6]. Although there are claims that primordial gravity
waves can be generated by other sources (e.g. phase transitions), such models
are more complicated and not widely accepted as a viable alternative.
1.1.1
Simple Single field inflation
It is worth presenting the simplest model of inflation (which became hugely relevant
after the latest BICEP2 data [6]), which is a model of a single inflaton field φ coupled
minimally to gravity with the potential V (φ) = 21 m2 φ2 .
The action for this model is
Z
S=
2
MPl
1 µν
1 2 2
d x −g
R − g ∂µ φ∂ν φ − m φ ,
2
2
2
4
√
(1.4)
leading to the following equation of motion
φ̈ + 3H φ̇ + m2 φ = 0 ,
(1.5)
where the Hubble parameter is defined as
1
H =
2
3MPl
2
1 2
φ̇ + V
2
.
(1.6)
During slow roll the dynamics is dominated by the potential energy and the Hubble
27
parameter remains almost constant, giving
1 2
φ̇ V (φ),
2
|Ḣ| H 2 ,
(1.7)
(1.8)
|φ̈| |3H φ̇| .
(1.9)
The above slow roll conditions are equivalent to 1 and |η| 1, where and η
are the slow-roll parameters defined by
2
V0
Ḣ
'− 2 ,
V
H
00 φ̈
V
2 ' −
|η| ≡ MPl
.
V H φ̇ 2
MPl
≡
2
(1.10)
(1.11)
During slow-roll we can neglect the φ̈ term in the equation of motion for the inflaton,
which gives
m2 φ
φ̇ = −
=−
3H
r
2
mMPl ,
3
(1.12)
where
H2 '
1
m2 2
V
=
φ .
2
2
3MPl
6MPl
(1.13)
Using this we can calculate the number of efolds of inflation (the number of times
that the universe expands by a factor of e ≈ 2.72)
Z
tend
dN = Hdt ⇒ N =
Z
φend
H(t)dt =
ti
φend
Z
=
φi
Z
φend
Hdt =
φi
2
−3H
dφ ⇒ N =
V0
φi
− φ2end
2
4MPl
φ2i
H
dt
dφ =
dφ
1 φ2i
⇒N ≈
. (1.14)
2
4 MPl
In order for inflation to solve the flatness and horizon problems we need Ne = 60 − 70.
1.1.2
Inflationary Perturbations
At this point we will present a simple description of the standard perturbation theory
for single field inflation. Since the thesis aims at extending these methods, this is a
28
good place to start. Since there are numerous books and review articles on inflationary
perturbations, we will not attempt to present a complete discussion here, but rather
a very simple introduction to make the comparison of the standard methods with our
new results easier.
We start by considering linear perturbations in the field and the metric
φ(x, t) = φ(x) + δφ(x, t),
gµν (x, t) = ḡµν (x, t) + hµν (x, t).
(1.15)
(1.16)
The line element in the longitudinal gauge in terms of the gauge-invariant Bardeen
potentials is
ds2 = −(1 + 2Φ)dt2 + a2 (1 − 2Ψ)dxi dxj δij .
(1.17)
In a single field model the anisotropic pressure vanishes, leading to the equality of
the two Bardeen potentials
Φ = Ψ.
(1.18)
The equations of motion for the field and metric perturbations are
1 2
∇ δφ + V,φφ δφ = −2ΨV,φ + 4φ̇Ψ̇ ,
a2
Ψ̇ + HΨ = 4πGφ̇δφ,
δ φ̈ + 3Hδ φ̇ −
(1.19)
(1.20)
supplemented by the constraint equation
1 2
Ḣ − 2 ∇ Ψ = 4πG(−φ̇δ φ̇ + φ̈δφ) .
a
(1.21)
Studying the evolution of perturbations becomes easier if one uses the gauge invariant
form
R=ψ+
29
H
δφ .
φ̇
(1.22)
The spectrum of the scalar perturbations defined by
k3
PR (k) ≡ 2
2π
Z
~
d3 xe−ik·~x hR(~x, t)R(~0, t)i.
(1.23)
Skipping the analysis, the spectrum of the scalar perturbations is
P(k) = As
k
k∗
ns −1
(1.24)
where the amplitude is calculated to be
As =
H2
2π φ̇
2
(1.25)
and the spectral tilt
ns = 1 − 6 + 2η .
(1.26)
Both As and ns are evaluated at the time that an arbitrary pivot wave number k∗
crosses the Hubble length (i.e., when k∗ /a = H). The Planck Collaboration uses
k∗ = 0.05 Mpc−1 . A similar analysis for the tensor modes gives for the ratio r of
tensor to scalar modes
r = 16 .
(1.27)
This means that the amplitude of tensor modes directly probes the value of the Hubble
parameter during inflation.
1.2
Beyond the single field
Modern theories of particle physics, including Supersymmetry and String Theory,
generally contain multiple scalar fields at high energies. Considering that some of
those scalars are relevant for inflation provides a model-building framework, where
one can deviate from the simple picture of a single slow rolling scalar field presented
above. And more than that multi-field models offer possibilities to extend beyond the
simplest case of purely adiabatic and Gaussian perturbations. They provide levers
30
with which to address some subtle features of state-of-the-art CMB measurements.
So there are both high-energy theory and empirical / phenomenological motivations
to look at multi-field models. We will study two classes of models, where a revision
or extension of the standard tools was needed to study inflationary perturbations.
1.2.1
Hybrid Inflation
As we saw for the single field inflation model, the same field is responsible for the expansion of the universe (since its potential energy dominates the Hubble parameter)
and the creation of the perturbations that are relevant for the CMB and structure formation. Inflation ends when the field starts accelerating near the end of the potential.
Hybrid inflation employs two fields with distinct roles.
1. The timer field produces the observable large scale perturbations.
2. The waterfall field provides the potential energy that drives the expansion and
once the timer field reaches a particular value, it is destabilized and ends inflation.
A pictorial representation of the hybrid potential is given in Fig. 1-1. The field rolls
along the valley (form right to left) and once it finds itself on top of the ridge is
destabilized by quantum perturbations and rolls towards its minimum. More details
regarding the transition are given in Chapter 2.
1.2.2
Non-minimal coupling
When we wrote the action for the simple model of inflation, the gravity term was
1
2
MPl
R.
2
However one can modify this, by essentially making Newton’s constant be
a function of the field amplitude at each point in space. In fact, renormalization of
scalar fields in a curved background generates terms of the form ξφ2 R [7, 8], where ξ is
a dimensionless parameter, called the non-minimal coupling. We will describe briefly
here the physics of a single non-minimally coupled inflaton field (for more look for
example [9, 10]), since in the main part of the thesis we study multiple non-minimally
31
Figure 1-1: A pictorial form of the hybrid potential.
coupled fields in detail. The general form of the action we will consider is
√
1 µν
d x −g f (φ)R − g ∂µ φ∂ν φ − V (φ) .
2
Z
4
S=
(1.28)
A conformal transformation is possible that will bring the gravity term to the usual
2
form of Einstein gravity 21 MPl
R
g̃µν =
2f
g .
2 µν
MPl
(1.29)
In addition we can make a field redefinition from φ to ϕ, in order for the kinetic term
to be canonical.
dϕ
MPl
=
dφ
2f (φ)
Ṽ
=
s
2f (φ) + 6(f 0 (φ))2
,
2f 2 (φ
V (φ)
.
(2f (φ))2
(1.30)
(1.31)
With the conformal transformation and the field redefinition the action becomes
Z
S=
2
p
MPl
1 µν
d x −g̃
R̃ − g̃ ∂µ ϕ∂ν ϕ − Ṽ .
2
2
4
(1.32)
In the case of multiple fields (more than two) the conformal transformation does not
change, but a field redefinition that makes the kinetic term canonical is not possible.
32
1.3
Organization of the thesis
The thesis is divided in two main parts, each consisting of a number of chapters and
dealing with two main classes of inflationary models. Each chapter is meant to be
self-contained and can be read separately. However Chapters 2 and 4 extensively
discuss the framework developed for the two model types that were studied and it
can be helpful to the reader to study them before proceeding to Chapters 3 and 5-7
respectively.
Chapters 2 and 3 deal with a new method of calculating density perturbations in
Hybrid Inflation. They are based on [11, 12] respectively. Regarding my contribution,
I did all numerical calculations and wrote most of the text for [11], and I assisted in
the analysis and wrote an early draft of [12].
Chapters 4-7 deal with calculating density perturbations in inflationary models
where multiple fields are coupled to each other and non-minimally to gravity. They
are based on [13, 14, 15, 16] respectively. Regarding my contribution, I did most
numerical calculations and a big part of the analytics and assisted in writing the final
version of the papers.
33
34
Bibliography
[1] A. H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon
and Flatness Problems,” Phys. Rev. D 23 (1981) 347.
[2] A. D. Linde, “A New Inflationary Universe Scenario: A Possible Solution of the
Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,”
Phys. Lett. B 108, 389 (1982).
[3] A. Albrecht and P. J. Steinhardt, “Cosmology for Grand Unified Theories with
Radiatively Induced Symmetry Breaking,” Phys. Rev. Lett. 48, 1220 (1982).
[4] P. A. R. Ade et al. [Planck Collaboration], “Planck 2013 results. XVI. Cosmological parameters,” arXiv:1303.5076 [astro-ph.CO].
[5] P. A. R. Ade et al. [Planck Collaboration], “Planck 2013 results. XXII. Constraints on inflation,” arXiv:1303.5082 [astro-ph.CO].
[6] P. A. R. Ade et al. [BICEP2 Collaboration], “BICEP2 I: Detection Of B-mode
Polarization at Degree Angular Scales,” arXiv:1403.3985 [astro-ph.CO].
[7] N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space (New York:
Cambridge University Press, 1982).
[8] I. L. Buchbinder, S. D. Odintsov, and I. L. Shapiro, Effective Action in Quantum
Gravity (New York: Taylor and Francis, 1992).
[9] D. I. Kaiser, “Primordial spectral indices from generalized Einstein theories,”
Phys. Rev. D 52, 4295 (1995) [astro-ph/9408044].
35
[10] A. Linde, M. Noorbala and A. Westphal, “Observational consequences of
chaotic inflation with nonminimal coupling to gravity,” JCAP 1103, 013 (2011)
[arXiv:1101.2652 [hep-th]].
[11] A. H. Guth and E. I. Sfakianakis, “Density Perturbations in Hybrid Inflation Using a Free Field Theory Time-Delay Approach,” arXiv:1210.8128 [astro-ph.CO]
[12] A. H. Guth, I. F. Halpern, M. P. Hertzberg, M. A. Joss and E. I. Sfakianakis,
“Hybrid Inflation with N Waterfall Fields: Density Perturbations and Constraints”, to appear.
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36
Chapter 2
Density Perturbations in Hybrid
Inflation Using a Free Field Theory
Time-Delay Approach
We introduce a new method for calculating density perturbations in hybrid inflation which avoids treating the fluctuations of the “waterfall” field as if they were
small perturbations about a classical trajectory. We quantize only the waterfall field,
treating it as a free quantum field with a time-dependent m2 , which evolves from
positive values to tachyonic values. Although this potential has no minimum, we
think it captures the important dynamics that occurs as m2 goes through zero, at
which time a large spike in the density perturbations is generated. We assume that
the time-delay formalism provides an accurate approximation to the density perturbations, and proceed to calculate the power spectrum of the time delay fluctuations.
While the evolution of the field is linear, the time delay is a nonlinear function to
which all modes contribute. Using the Gaussian probability distribution of the mode
amplitudes, we express the time-delay power spectrum as an integral which can be
carried out numerically. We use this method to calculate numerically the spectrum of
density perturbations created in hybrid inflation models for a wide range of parameters. A characteristic of the spectrum is the appearance of a spike at small length
scales, which can be used to relate the model parameters to observational data. It is
37
conceivable that this spike could seed the formation of black holes that can evolve to
become the supermassive black holes found at the centers of galaxies.
2.1
Introduction
Inflation [1] remains the leading paradigm for the very early universe. It naturally
solves the cosmological flatness and horizon problems and is consistent with high precision measurements of the cosmic microwave background radiation [2, 3]. Numerous
models of inflation have been proposed, each adding features to the predictions of a
scale invariant spectrum derived from single-field slow-roll inflation. Their motivation can be either some particle physics ideas coming from the standard model [4] or
supersymmetric theories [5, 6], the need to explain some observation such as glitches
in the CMB or supermassive black holes in galactic centers, or simply the extension
of a theorist’s toolbox in anticipation of the next set of high precision data, such as
the upcoming Planck satellite measurements.
Hybrid inflation was first proposed by A. Linde [7] and the name was chosen
because this class of models can be thought of as being a hybrid between chaotic
inflation and inflation in a theory with spontaneous symmetry breaking. The simplest
hybrid inflation model requires two fields that we will call the timer and waterfall
fields. The timer field corresponds the usual slow rolling field and is responsible for
the scale invariant spectrum of perturbations observed in the CMB. The waterfall field
is confined to its origin by the interaction with the timer field, giving a large constant
contribution to the potential, which is also the main contribution to the energy density
and hence the Hubble parameter. The potential governing the waterfall field changes
as the timer field evolves, and at some point the minimum of the potential turns into
a local maximum, and the waterfall field rolls down its tachyonic potential to its new
minimum, where inflation ends. A characteristic feature of the density perturbation
spectrum of hybrid inflation is the appearance of a large spike generated at the time
when the waterfall potential turns tachyonic. The spike is generically at small length
scales, and can potentially seed primordial black holes [8]. Primordial black hole
38
formation and evolution has been studied in the past [9, 10, 11, 12], but whether
these black holes grow to become the supermassive black holes currently found in
galactic centers is an open and intriguing possibility that we will address in a future
publication.
Usual inflationary perturbation theory is based on the study of quantum fluctuations around a classical trajectory in field space. However, in a purely classical
formulation the waterfall field of hybrid inflation would remain forever at the origin,
even after the waterfall transition, due to symmetry. It is quantum fluctuations that
destabilize it and lead to the end of inflation, so in a sense the classical trajectory
has a quantum origin. Numerous papers have used various analytical approaches or
numerical simulations to overcome this difficulty and approximate the spectrum of
density perturbations [5, 6, 15, 16, 17, 18, 19, 20, 21, 13, 14].
The method we use here has evolved from the early work in Kristin Burgess’ thesis
[13], in which she studied a free-field model of the waterfall field in one space dimension, focusing on the time delay of the scalar field as a measure of perturbations. As
in the model considered here, the waterfall field was described by a Lagrangian with
a time-dependent m2 , caused by the interaction with the timer field. m2 evolved from
positive values at early times to negative (tachyonic) values at late times. Such models
are unnatural, since the potential is not bounded from below, but they nonetheless
appear to be useful toy models, since the dynamics that generate the spike in the
fluctuation spectrum occur during the transition from positive to negative m2 . The
evolution in the bottomless potential is realistic enough to give a well-defined time
delay. Burgess studied the evolution of the waterfall field by means of a numerical
simulation on a spatial lattice, using 262,000 points, calculating the power spectrum
of the time delay by Monte Carlo methods. The method was slow, but for one choice
of parameters she accumulated 5000 runs, giving a very reliable graph of the timedelay spectrum for this model. This line of research was pursued further in the thesis
of Nguyen Thanh Son [14], who repeated Burgess’ numerical simulations with a new
code (with excellent agreement). More importantly, Son and one of us (AHG), with
some crucial input from private communication with Larry Guth, developed a method
39
to short-circuit the Monte Carlo calculation. Instead of determining the power spectrum by repeated random trials, it was possible to express the expectation value for
the random trials as an explicit expression involving integrals over mode functions,
which could be evaluated numerically. The speed and numerical precision were dramatically improved. While Son’s work was still limited to one spatial dimension, the
possibility of extending it to three spatial dimensions was now a very realistic goal. In
this chapter we extend the calculation of the time-delay power spectrum in free-field
models of hybrid inflation to three spatial dimensions, calculating the spectrum for a
wide range of model parameters.
In Section II we define the free-field model for the timer and waterfall fields that
we will use to calculate fluctuations. We set up the equations of motion, define the
notation of the mode expansion, and discuss the behavior of the mode functions. We
make contact with a class of supersymmetric models that support hybrid inflation
in Section III, presenting the form of their potential and the range of parameters
that they allow. Section IV gives a brief summary of the time delay formalism, and
presents an approximation for calculating perturbations, developed earlier by Randall,
Soljačić, and Guth. In Section V we develop a new method for calculating density
perturbations in hybrid inflation that avoids any need to consider small fluctuations
about a classical solution. Instead we show how the time delay power spectrum can
be calculated essentially exactly in the context of the free field theory description.
The result is given in the form of an integral over the modes which makes use of their
known Gaussian probability distribution. In section VI we present an extensive set of
numerical results over the parameter space of our model, where we are able to isolate
the main factors that influence the density perturbation spectrum. In the limit of a
light timer field, all quantities of interest are determined by the product of the timer
and waterfall masses. We examine the models discussed in Section II as examples
of realistic versions of hybrid inflation, and provide graphs showing the predictions
of these models. Concluding remarks and directions of future work follow in Section
VII.
40
2.2
Model
As we noted in the Introduction, the model consists of two fields with different behavior: the timer and waterfall fields. The timer field dominates the initial slow-roll
inflation phase and we take it to be a single scalar field. The waterfall field becomes
tachyonic and rolls to its minimum ending inflation. We take the waterfall field to
be a complex scalar, so as to avoid the creation of domain walls after the waterfall
transition. The action of this two-field system with minimal coupling to gravity is
Z
S=
2
MPl
1 µν
1 µν
†
R − g ∂µ ψ∂µ ψ − g ∂µ φ ∂ν φ − V (ψ, φ) ,
d x −g
2
2
2
4
√
(2.1)
where the potential is comprised of a constant term, the potentials of the waterfall
and timer fields and an interaction terms.
V (ψ, φ) = V0 + Vψ (ψ) + Vφ (φ) + Vint (ψ, φ).
2.2.1
(2.2)
Field set-up
Our first assumption is related to the expansion rate. We consider the metric to be
exactly De-Sitter, even though this is only approximately correct. However, it changes
only weakly during the slow roll inflation era, and we will terminate our calculation
once the approximation loses its validity. This is equivalent to requiring the constant
term V0 to dominate the potential. Defining the Hubble constant during inflation
through
V0
,
3Mpl2
(2.3)
a(t) = eHt .
(2.4)
H2 ≈
the scale factor is written as
Our second approximation concerns the potential of the two fields Vψ (ψ) and Vφ (φ).
We will take both to be purely quadratic. Physically this can only be an accurate
approximation close to the transition time. However we will take this potential to
hold for all times. Since the important dynamics happens close to the time of the
41
transition, we believe that this approximation makes the dynamics tractable without
introducing large errors. Work is already under way to extend out method to more
realistic potentials.
We will split the action in two parts for each of the fields. We neglect the interaction term from the Lagrangian of the timer field and consider it a part of the waterfall
field Lagrangian. This means that there is no back-reaction from the waterfall to the
timer field. Physically this is a reasonable approximation before the waterfall transition, as well as afterwards, for as long as the waterfall field remains close to the
origin. Mathematically, neglecting this term makes the equation of motion for the
timer field de-coupled and in our quadratic approximation analytically solvable.
The Lagrangian density for the waterfall field φ is
3Ht
Lφ = e
h
2
−2Ht
|φ̇| − e
2
|∇φ| −
m2φ (t)|φ|2
i
.
√1 (φ1
2
The usual 1/2 factors can be restored, if one writes φ =
(2.5)
+ iφ2 ) where φ1 and
φ2 are real scalar fields. The waterfall field must be complex, otherwise it will create
domain walls as it rolls down from its initial value. The time-dependent mass of the
φ field is controlled by a real scalar field, subsequently called the timer field. The
important property of the squared mass of φ is that it has to be positive initially and
as ψ evolves become negative. A general form is the following
m2φ (t)
=
−m20
1−
ψ(t)
ψc
r .
(2.6)
We will choose r = 4 for most of our simulations. The lagrangian density of the timer
field ψ is
Lψ = e
3Ht
1 2 1 −2Ht
1 2 2
2
ψ̇ − e
(∇ψ) − mψ ψ .
2
2
2
(2.7)
We do not examine perturbations arising from quantum fluctuations of the timer
field. Before the waterfall transition they will give the nearly scale invariant spectrum
that can be matched to the CMB observations. Apart from making sure that the
long wavelength tail of the waterfall field perturbations does not contradict CMB
42
data, we will not consider these scales. After the waterfall transition the timer field
perturbations will continue to be of the order of 10−5 , hence they will be subdominant
to the perturbations of the waterfall field by a few orders of magnitude, as we will see.
We will not worry about matching the Plack observations for the spectral index of the
CMB, but we will restrict ourselves to a quadratic timer field potential. Comments
on the modification needed to turn the blue spectrum of this model to one compatible
with Planck are deferred for [25].
The equations of motion are
φ̈ + 3H φ̇ − e−2Ht ∇2 φ = −m2φ (t)φ,
(2.8)
ψ̈ + 3H ψ̇ − e−2Ht ∇2 ψ = −m2ψ ψ.
(2.9)
If we take the timer field to be spatially homogenous, we get
ψ(t) = ψc ept
(2.10)
with the exponent


s
3
p± = H − ±
2
9
−
4
m2ψ
.
H2
(2.11)
The value of the constant of integration was chosen so that ψ(t) = ψc and m2φ (t) = 0
at t = 0. Both roots are negative, but the long time behavior is dominated by the
larger of the two roots, which is

3
p = p+ = −H  −
2
We will always choose
mψ
H
<
3
2

s
9
−
4
m2ψ
.
H2
(2.12)
and not consider the case of a complex root. In fact,
hybrid inflation models usually require the mass of the timer field to be well below
the Hubble parameter, as in [5] and [6]. We choose to measure time in number of
e-folds, hence we use N = Ht. We rescale the masses similarly as µψ = mψ /H and
µφ = m0 /H. Furthermore the finite box size that we will use in our simulations is
43
measured in units of
1
H
and the field magnitude in units of H. We also define

µ̃2ψ = −

s
rp
3
= r −
H
2
9
−
4
m2ψ
.
H2
(2.13)
For a light timer field the reduced mass µ̃ψ is proportional to the actual timer mass,
p m p
µ̃ψ = 3r Hψ = 3r µψ .
2.2.2
Fast Transition
Let’s consider the speed of the transition. The transition happens at m2φ = 0. In order
to quantify the speed of the transition, we will use the basic scale of our system, the
Hubble scale. We will consider the transition duration to be the period for which
|mφ | ≤ H, meaning that the mass term in the equation of motion of the waterfall
field is negligible. Assuming that µ̃φ > 1 we get
±1 =
µ2φ
−µ̃2ψ N
1−e
1
⇒ ∆N = 2 log
µ̃ψ
µ2φ + 1
µ2φ − 1
!
.
(2.14)
In the limit of µ̃φ 1
∆N =
2
.
(µ̃ψ µφ )2
(2.15)
Another measure of the speed of the transition is given by derivative of the waterfall
field mass at N = 0.
1 dm2φ (N )
1
.
|N =0 = (µ̃ψ µφ )2 ⇒ ∆N ∼
2
H
dN
(µ̃ψ µφ )2
(2.16)
This shows that as long as the product µφ µψ is somewhat larger than unity, the
duration of the transition will be less than a Hubble time, meaning that the transition
is fast!
44
2.2.3
Mode expansion
For purposes of our numerical calculations, we think of the universe as a finite box
with periodic boundary conditions and a discrete spatial lattice. We choose the lattice
to be cubic with length b and Q3 points. This means that
2π
b~
l , ~k =
~n ,
Q
b
~x =
(2.17)
where ~l is a triplet of integers between 0 and Q − 1 and ~n is a triplet of integers
between −Q/2 and (Q/2) − 1. We can move between the finite discrete set of points
and the continuous limit using the usual substitutions
Z
3 X
3 X
Z
b
2π
3
d x→
,
dk→
.
Q
b
3
~
x
(2.18)
~k
Our convention for the Fourier transform is
Z
f (~x) =
f (~k) =
3
i~k·~
x
d ke
1
2π
3 Z
f (~k) =
i~k·~
x
3
d xe
2π
b
3 X
~k
f (~x) =
~
eik·~x f (~k),
1
2π
3 3 X
b
~
eik·~x f (~x) .
Q
(2.19)
~
x
We will expand the waterfall field in modes in momentum space,
1
φ(~x, t) =
(2π)3/2
2π
b
3/2 X
~
~
[c(~k)eik·~x u(~k, t) + d† (~k)e−ik·~x u∗ (~k, t)] ,
(2.20)
~k
which with Eq. (2.8) gives
2
ü(~k, N ) + 3u̇(~k, N ) + e−2N k̃ 2 u(~k, N ) = µ2φ (1 − e−µ̃ψ N )u(~k, N ) ,
where k̃ =
|~k|
H
(2.21)
and an overdot denotes a derivative with respect to the time variable
N = Ht.
45
2.2.4
Solution of the mode function
Early time behavior
At asymptotically early times the k̃ 2 term dominates over the mass term provided
√
that µ̃2ψ < 2. For r = 2 this is the case for µψ < 2, and for r = 4 it holds for
p
µψ < 5/4. These inequalities will hold throughout the parameter space of the
models that we will examine, so we can neglect the mass term for N → −∞. We
then define a new function, following [22] as
1
u(~k, N ) =
2
r
π −3N/2
e
Z(z) , z = k̃e−N .
H
(2.22)
Neglecting the mass term in Eq. (2.21), we find
2
Z
dZ
9
2
z
+z
+ z −
Z=0,
dz 2
dz
4
2d
(2.23)
which is the equation for a Bessel function of order 3/2. At early times the solution
should look like a harmonic oscillator in its ground state, or equivalently the ground
state of a free field in flat space, which is composed of negative frequency complex exponentials. This choice of initial conditions is the well known Bunch–Davies vacuum.
For a review of scalar field quantization in de Sitter space and the corresponding
vacuum choice see for example Ref. [23]. At early times the solution is given by
1
u∼
2
r
π −3N/2 (1)
e
H3/2 (z) ,
H
(2.24)
where
(1)
H3/2 (z)
r
=−
2 iz
1
e
1−
πz
iz
(2.25)
is a Hankel function, a linear combination of Bessel functions. (The phase is arbitrary, and the normalization is fixed by insisting that the field and the creation and
annihilation operators obey their standard commutation relations.) Rewriting the
46
original mode equation in terms of the new variable z, it simplifies to
"
µ̃2ψ #
µ2φ
∂ 2 u 2 ∂u
z
+u= 2 1−
−
u.
2
∂z
z ∂z
z
k̃
(2.26)
The ~k = 0 mode is not captured by the procedure described here and is presented in
detail in Appendix 2.9.1.
General Solution
We will now examine the general solution in a form that will be more appropriate for
the numerical calculations that we have to perform. We can write the solution as
1
~
R(~k, t)eiθ(k,t)
u(~k, t) = p
2k̃H
(2.27)
and the differential equation separates in real and imaginary parts
2
R̈ − Rθ̇2 + 3Ṙ + e−2N k̃ 2 R = µ2φ (1 − e−µ̃ψ N )R,
2Ṙθ̇ + Rθ̈ + 3Rθ̇ = 0.
(2.28)
(2.29)
Integrating the second equation gives
e−3N
θ̇ = const 2 .
R
(2.30)
By comparing this with the early time behavior of the analytic solution
u∼
1
−N
p e−N eik̃e .
2H k̃
(2.31)
The phase equation becomes
θ̇ = −
k̃e−3N
,
R2
47
(2.32)
while the initial condition for the amplitude is given by the same asymptotic term to
be
R → e−N .
(2.33)
Inserting this expression in the equation for the amplitude function R
R̈ −
k̃ 2 e−6N
2
+ 3Ṙ + e−2N k̃ 2 R = µ2φ (1 − e−µ̃ψ N )R .
3
R
(2.34)
A closer look at the mode behavior
Let us rewrite the equation of motion (Eq. 2.21) in a way that makes the time
dependence of the solution more transparent
2
ük (t) + 3u̇k (t) + µ2ef f = 0 , µ2ef f (k) = k̃ 2 e−2N + µ2φ e−µ̃ψ N − µ2φ .
(2.35)
We can distinguish different time windows with different behavior of the mode functions, based on the effective waterfall field mass. We will list these time windows here
and then proceed to examine them one by one.
1. N 0, many efolds before the waterfall transition, in the asymptotic past
2. Ndev (k) < N < 0, a few efolds before the transition, where Ndev (k) is the time at
which a mode starts deviating significantly from the e−N behavior, in particular
starts decaying faster.
3. 0 < N < Ntr (k) a few efolds after the transition, where Ntr (k) is the time at
which each mode starts growing.
4. N 0, the asymptotic future
Now let us look at each of those time scales more closely. The asymptotic past is
well described in the previous section and we see that all modes decay like e−N .
q
1 −N
More precisely their magnitude behaves as |uk | ∼ 2k
e . The first time scale Ndev
appears only for low wavenumbers. For N < 0 we can keep only two of the three
2
terms in the effective mass. Since µ2φ e−µ̃ψ N > µ2φ we will drop the µ2φ term, leaving the
48
2
effective mass as µ2ef f (k) = k̃ 2 e−2N + µ2φ e−µ̃ψ N . The time at which the two dominant
terms become equal is
2
Ndev (k) =
log
2 − µ̃2ψ
k̃
µφ
!
.
(2.36)
For k̃ ≥ µφ this time is not negative, hence we cannot drop µ2φ and our analysis fails.
This transition, which happens only for k̃ < µφ signals a deviation of the behavior of
the modes, which do not evolve as e−N , but instead decay faster.
Next we move to the actual waterfall transition time for each mode, which happens
when the effective squared mass changes sign and becomes negative, or
2 −2N
k̃ e
=
µ2φ
−µ̃2ψ N
1−e
.
(2.37)
We will approximate the right hand side of the above equation with a piecewise linear
function as follows
2
µ2φ 1 − e−µ̃ψ N

 µ2 µ̃2 N : N < 1/µ2
ψ
φ ψ
.
=
 µ2
2
:
N
>
1/µ
φ
ψ
(2.38)
2
For k̃ < µφ e1/µψ the solution is found on the first branch and is
1
Ntr (k) = W
2
2k̃ 2
µφ µ̃2ψ
!
.
(2.39)
where W is known as the Product Logarithm, or Lambert W function and is defined
as the solution to the equation z = W (z)eW (z) . For small values of the wavenumber
we can write the solutions as a Taylor series in k̃
Ntr (k̃ p
µφ µ̃ψ ) =
k̃ 2
+ O(k̃ 4 ).
2 2
µφ µ̃ψ
(2.40)
2
For k̃ > µφ e1/µψ we operate on the second branch and the transition time for each
mode is
Ntr (k) = log
49
k̃
µφ
!
.
(2.41)
The behavior of the modes after the transition is different for different ranges of
the timer field mass. If we consider the late time behavior of the mode equation,
we can see two timescales introduced by the time dependent exponential coefficients.
One is O(1) and the other O(1/µ̃2ψ ). We distinguish two cases: They can both be
O(1) or the second one can be larger than the first. The first timescale defines the
time at which the equation becomes k-independent, meaning that all modes behave
(grow) in the same way. The second time scale defines the time, after which the
equation becomes time independent, meaning that after that all modes behave as
pure exponentials.
Let us first deal with the case of µ̃ψ 1 meaning that the second time scale is
much larger than the first one. Between the two timescales, that is 1 < N < −1/µ̃2ψ ,
the equation is independent of k̃
2
R̈ + 3Ṙ = µ2φ (1 − e−µ̃ψ N )R.
(2.42)
Since the evolution of the exponential term on the right hand side is by far slowest
2
than the other timescales in the problem, we will treat 1 − e−µ̃ψ N adiabatically. This
leads immediately to the solution
RN
R = R0 e
0
λ(N 0 )dN 0
, λ(N ) =
q
2
−3 + 9 + 4µ2φ (1 − e−µ̃ψ N )
2
.
(2.43)
After a long time the growth rate would mathematically settle to
1 dR
→ λ∞ =
R dN
−3 +
q
9 + 4µ2φ
2
.
(2.44)
However, this is far beyond the time when inflation will have ended, hence it would
physically never have time to happen (plus it is well outside the validity of our constructed potential).
Let us choose µφ = 10 and µ̃ψ = 1/10 to demonstrate our analysis. Some characteristic mode functions are presented in Fig. 2-1
50
Figure 2-1: Mode functions for different comoving wavenumbers as a function of
1
and µφ = 10. We can see the
time in efolds. The model parameters are µψ = 10
modes following our analytic approximation for the growth rate. Our analysis gives
Ndev (1/256) ≈ −7.9 and Ntr (256) ≈ 4.76, which are very close to the values that can
be read off the graph.
We see that all modes behave similarly at late times, independent of their wavenumber, as they should based on our late time analysis. Specifically, we can plot the
ratio of the time derivative of each mode to its magnitude, as in Fig. 2-1. We call
k
this the growth rate λ ≡ Ṙ
. We can see both phenomena. First, after N ≈ 6
Rk
the modes behave identically. Second, the behavior of the mode approaches that of
an exponential function (whose logarithm is a constant), but at a slower rate. In
this example time needs to go on for several hundreds of efolds for the growth rate
to set to a constant, which is calculated to be λ(t → ∞) = 8.6119 for µφ = 10 and
µψ = 1/10.
It is important to test our analytical approach to the late time behavior of the
growth rate of mode function. As seen in Fig. 2-1 once the mode functions evolve in
a k - independent way, our simple analytical estimate for their growth rate is accurate
to within a few percent, which gives us a very accurate expression for the growth rate
and leads to the terms evolving as u ∼ eλ(t)t , where the time-dependent growth rate
λ(t) is slowly changing.
As a test of our analysis, we can calculate the two important transition times
Ndev (1/256) = −7.9005 and Ntr (256) = 4.76457. We see that the calculated values
51
Figure 2-2: Mode functions for different comoving wavenumbers as a function of
time in efolds. The model parameters are µψ = 21 and µφ = 1. The horizontal line
corresponds to the asymptotic value of the growth factor λ. We can see how the
mode functions reach their asymptotic behavior after 10 efolds. Our analysis gives
Ndev (1/256) ≈ −7.84 and Ntr (256) ≈ 5.4, which are very close to the values that can
be read off the graph.
agree very well with the behavior of the plotted modes.
Let us briefly examine the situation where µ̃2ψ ≤ 2. In this case the mode equation
becomes k-independent and time independent at about the same time, that is a few
e-folds after the waterfall transition. We choose µψ =
1
2
and µφ = 1 and plot the
results in Fig. 2-2. It is clear that the modes become both k-independent and pure
exponential (having a constant growth rate) at about the same time (N ≈ 10). The
asymptotic growth rate in this case is λ(t → ∞) = 0.3028.
Again we can calculate the two important transition times Ndev (1/256) = −7.8377
and Ntr (256) = 5.39554, which agree once more with the behavior of the plotted
modes.
2.3
Supernatural Inflation models
It is interesting to make contact between our abstract model and specific potentials
inspired form particle theory. In general inflation models require small parameters
in order to ensure slow roll inflation and produce the correct magnitude of density
perturbations. It was shown in [5, 6] that supersymmetric theories with weak scale
52
Quadratic Approximation
SUSY Model 1
V0
M4
SUSY Model 2
M4
m0
2
M
√
2f
M2
√
2f
r
4
2
ψ
√c
M √2M 0
√ f2
2M
fλ
mψ
mψ
mψ
Table 2.1: Parameters of SUSY models and their counterparts in our quadratic approximation
supersymmetry breaking can give models where such small parameters emerge ”naturally” as ratios of masses already in the theory. We will not go into the details of
such theories, but instead give the forms of the constructed potentials and use them
as an application of our formalism.
√
m2ψ 2 ψ 4 φ2 + φ4 ψ 2
V = M 4 cos2 (φ/ 2f ) +
ψ +
2
8M 0 2
(2.45)
for what we will call model 1 and will be the primary focus of this work and
√
m2ψ 2
ψ 2 φ2
ψ + λ2
V = M 4 cos2 (φ/ 2f ) +
2
4
(2.46)
which we will call model 2.
The first model can be taken with M 0 at one of three regions: the Planck scale,
the GUT scale or an intermediate scale (∼ 1010 GeV ). At each scale the rest of the
parameters are adjusted accordingly to produce sufficient inflation and agree with
CMB data.
We will approximate the potential with a pure quadratic one with a time varying
waterfall mass, of the form
V (φ, ψ) = V0 −
m20
1−
ψ
ψc
r |φ|2 + m2ψ ψ 2 ,
(2.47)
where r = 4 for model 1 and r = 2 for model 2, as can be easily seen by the form
of the interaction terms in both cases. The correspondence between the exact SUSY
potential and our quadratic counterpart is shown in Table 2.1.
The parameters of the two models are restricted to fit CMB data, as shown in
Fig. 2-3.
53
Figure 2-3: Parameter space for the two supernatural inflation models. The bottom
right corner shows the parameter for model 2, while the other three show parameters
for model 1, for different ranges of the mass scale M 0
54
To lowest order, in this potential dominated model, the Hubble parameter is
constant and equal to
r
H=
2.3.1
8π M 2
=
3 Mp
r
√
8π V0
.
3 Mp
(2.48)
End of Inflation
In our simplified quadratic model inflation will never end. The waterfall field will
roll forever down its tachyonic potential. However, we shall not forget that this
is a mere Taylor expansion of more realistic potentials, which have a well defined
minimum. We will use the supersymmetric potentials of [5, 6] as a concrete example
to connect our purely quadratic potential to ones with more realistic shapes. In these
supersymmetric models the potential has a cosine-like form and the minimum occurs
at
√φ
2f
= π2 , where the inflaton will oscillate, terminating inflation and giving rise to
(p)reheating. By making contact between the parameters of our potential and the
physical parameters of the actual supersymmetric models, we can estimate the field
value at which inflation ends.
There are two strategies for defining φend , the field value at which inflation ends.
We can either pretend that the quadratic potential can be followed up to the end field
value of the corresponding SUSY potential, or we can choose to end our calculation
when the quadratic potential departs significantly from the actual SUSY potential
that we are trying to approximate.
√
In the first case the end field value is at φend = f π/ 2. To calculate the end
field value for the latter case we will note that the cosine potential is accurately
approximated by a quadratic as long as φ/f 1. We will call this ratio and in this
case we will end our calculations when ceases being small. We can write these two
cases in a unified manner, as
φend = f,
(2.49)
√
where = π/ 2 if we follow the quadratic potential all the way to the field value
corresponding to the minimum of the SUSY potential and < 1 if we stop our
calculation at the point where the quadratic potential deviates significantly from the
55
supersymmetric one.
Using the values of the parameters taken from the supersymmetric models, we
can estimate the end field value to be
φend ∼ 1015 H
(2.50)
within one or two orders of magnitude for all cases of models considered in [5, 6].
We will be using field values of this order of magnitude in our numerical calculations, whether we are dealing with the supersymmetric potentials or not. We will
however examine the effects of changing the end value of the field and show that it is
minimal, easily understandable, and calculable.
2.4
Perturbation theory basics
2.4.1
Time delay formalism
The time delay formalism provides an intuitive and straightforward way to calculate
primordial perturbations. Its basic principle is that inflation ends at different places in
time at different times, due to quantum fluctuations. This leads some of the regions of
the universe to have inflated more than others, creating a difference in their densities.
The time-delay formalism was first introduced by Hawking [24] and by Guth and Pi
[26], and has recently been reviewed in Ref. [27].
We will briefly describe the method here for the case of a single real scalar field.
The universe is assumed to be described by a de-Sitter space-time, since the Hubble
parameter is taken to be a constant. The equation of motion for the scalar field φ(~x, t)
is
φ̈ + 3H φ̇ = −
1
∂V
∇2 φ,
+
2
∂φ
a(t)
(2.51)
where the last term is suppressed by an exponentially growing quantity, so at late
times it becomes negligible. We will omit the last term from now on.
We call the homogenous (classical) solution φ0 (t) and write the full solution, in56
cluding a space dependent small perturbation δφ φ0 as
φ(~x, t) = φ0 (t) + δφ(~x, t).
(2.52)
Plugging this into the equation of motion and working to linear order in δφ one can
show that the quantity δφ obeys the same differential equation as φ̇0 . Furthermore
the presence of a damping term implies that any two solutions approach a time
independent ratio at large times. Thus, at large times we have (to first order in δτ )
δφ(~x, t) → −δτ (~x)φ̇0 (t) ⇒ φ(~x, t) → φ0 (t − δτ (~x)).
(2.53)
This is the formulation of the intuitive picture of the time delay method.
2.4.2
Randall-Soljacic-Guth approximation
The usual calculation of density perturbations in inflation considers small quantum
fluctuations around a classical field trajectory. In the case of hybrid inflation such
a classical trajectory does not exist, since classically the field would stay forever on
the top of the inverted potential. It is the quantum fluctuations that push the field
away from this point of unstable equilibrium. One way to overcome this difficulty
is to consider the RMS value of the field as the classical trajectory. This was done
for example in [5] and [6] and is a recurring approximation in the study of hybrid
inflation.
Using the Bunch-Davies vacuum in the definition of the RMS value of the waterfall
p
field φrms = h0|φ(x, t)φ∗ (x, t)|0i it it straightforward to calculate it using the mode
expansion
φ2rms (t) =
E
1 X i(~k−~k0 )·x D
1 X
†
† ∗
∗
e
0|(c
u
+
d
u
)(c
u
+
d
u
)|0
=
|uk (t)|2 .
~k ~k
~k0 ~k0
−~k0 −~k0
3
−~k −~k
b3
b
0
~k , ~k
~k
(2.54)
57
The mean fluctuations are measured by
∆φ(~k) =
"
k
2π
3 Z
#1/2
i~k·~
x
d3 xe
hφ(x)φ∗ (0)i
"
=
k
2π
3
#1/2
|u~k |2
(2.55)
resulting in what will be called the RSG approximation for the time delay field
pP
2
~k, t) kb 3/2
∆φ(
~ |u~ (t)|
|u~k | P k k
.
∆τRSG (~k) ≈
=
2π
φ̇rms
~k u̇~k (t)u~k (t)
(2.56)
There is an important comment to be made about the quantum mechanical nature
of these density perturbations. In regular models of inflation quantum perturbations
are scaled by ~. We can think of them as modes with initial conditions that are of
the order of ~. The classical trajectory on the other hand does not have any quantum
mechanical origin, hence does not scale with ~. This means that in the limit of ~ → 0
the perturbations vanish, as one would expect will happen if one could ”switch off”
quantum mechanical effects.
In the case of hybrid inflation on the other hand, what we call the classical trajectory (be it the RMS value or something else) is comprised of modes that originated
as quantum fluctuations, hence is scaled by ~ itself. This means that even in the limit
of ~ → 0, the density perturbations in hybrid inflation remain finite! By explicitly
restoring ~ in the formulas of this chapter, the reader can formally arrive to the same
conclusion.
Some plots of the time delay field calculated using the RSG approximation are
shown in Fig. 2-4. The reduced mass of the timer field was taken to be µψ =
1
20
while
we varied the waterfall field mass. We fixed the time at which inflation ended to be
15 e-folds after the waterfall transition.
58
Figure 2-4: The time delay field calculated using the RSG formalism. The end time
1
was taken to be 15 e-folds after the waterfall transition and µψ = 20
for all curves,
while we varied µφ .
2.5
Calculation of the Time Delay Power Spectrum
The usual method to calculate the primordial perturbation spectrum would involve
either making some approximations (more or less similar to the RSG) or using a
Monte Carlo simulation. The former suffers from the lack of a classical trajectory that
invalidates the usual perturbation method, while the latter would be computationally
costly in three spatial dimensions. We will therefore implement an alternate method
that reduces the calculation of the spectrum of the time delay field to the evaluation
of a two dimensional integral and does not need a classical trajectory to do so.
As discussed at the end of Sec. 2.2.4, the behavior of the mode functions at
asymptotically late times (t → ∞) is given by
u(~k, t → ∞) ∼ eλ∞ t u(~k) ,
(2.57)
where λ∞ is given by Eq. (2.44). If we define for all times
P R(~k,t)Ṙ(~k,t)
~k
φ̇rms (t)
2|k|
λ(t) ≡
= P
,
|R(~k,t)|2
φrms (t)
~
k
59
2|k|
(2.58)
then at late times λ(t) → λ∞ . Since λ(t) changes very slowly, we can take it as a
constant around the time of interest.
To discuss fluctuations in the time at which inflation ends, we begin by defining
t0 as the time when the rms field reaches the value φend , which we have chosen to
define the nominal end of inflation:
φ2rms (t0 ) = φ2end .
(2.59)
Since at late times all modes, to a good approximation, grow at the same exponential
rate λ(t), we can express the field φ(~x, t) at time t = t0 + δt in terms of the field
φ(~x, t0 ) by
|φ(~x, t)|2 = |φ(~x, t0 )|2 e2
R t0 +δt
t0
λ(t0 )dt0
= |φ(~x, t0 )|2 e2λ(t0 )δt .
(2.60)
We will drop the argument of λ(t0 ) from now on. If t is chosen to be the time tend (~x)
at which inflation ends at each point in space, then φ ~x, tend (~x) = φend = φrms (t0 ),
and the above equation becomes
φ2rms (t0 ) = |φ(~x, t0 )|2 e2λδt ,
(2.61)
which can be solved for the time delay field δt(~x) = tend (~x) − t0 :
−1
δt(~x) =
log
2λ
|φ(~x, t0 )|2
φ2rms (t0 )
.
(2.62)
Rescaling by the rms field
φ(~x, t)
,
φrms (t)
(2.63)
−1
log |φ̃(~x, t0 )|2 .
2λ
(2.64)
φ̃(~x, t) ≡
we can write
δt(~x) =
Using this expression, we can write the two-point function of the time delay field as
D
E
E
1 D
δt(~x)δt(~0) = 2 log |φ̃(~x, t0 )|2 log |φ̃(~0, t0 )|2 ,
4λ
60
(2.65)
which can be evaluated, since the probability distributions are known. To continue,
we can decompose the complex scalar field in terms of the real fields Xi :
φ̃(~x, t) = X1 + iX2 , φ̃(~0, t) = X3 + iX4 .
(2.66)
The average value of a function F of a random variable X with probability distribution
function p(X) is given by
Z
hF [X]i =
dXp(X)F [X] .
(2.67)
Since this is a free field theory, we can take the four random variables Xi (~x) to follow
a joint Gaussian distribution with
1
1 T −1
p
p(X) =
exp − X Σ X
, Σij = hXi Xj i .
2
(2π)2 det(Σ)
(2.68)
A function of the Xi0 s then has the expected value
hF [X]i =
Z Y
4
1
1 T −1
p
dXi
exp − X Σ X F [X] .
2
2
(2π)
det(Σ)
i=1
(2.69)
The new fields Xi can be written in terms of the original complex field φ as
X1
X3
i
i
1 h
1h
∗
∗
φ̃(~x) + φ̃ (~x) , X2 =
φ̃(~x) − φ̃ (~x) ,
=
2
2i
h
i
h
i
1
1
∗ ~
∗ ~
~
~
=
φ̃(0) + φ̃ (0) , X4 =
φ̃(0) − φ̃ (0) .
2
2i
(2.70)
The components of the variance matrix Σ can be easily calculated using the commutation relations for the creation and annihilation operators in φ(~x, t), from Eq. (2.20).
61
Due to the high degree of symmetry the matrix itself has a very simple structure:


1
2
0 ∆ 0




1
 0 2 0 ∆ 
 ,
Σ=


 ∆ 0 12 0 


1
0 ∆ 0 2
(2.71)
where
∆(~x, t0 ) = hX1 X3 i = hX2 X4 i =
E
1 X ~
1D ∗
~
φ (~x, t0 )φ(~0, t0 ) = 3
|ũ(k, t0 )|2 eik·~x ,
2
2b
~k
(2.72)
where
u(~k, t)
ũ(~k, t) =
.
φrms (t)
(2.73)
Since ũ(~k, t) actually depends only on the magnitude of the wavenumber, because of
the isotropy of the problem, we can do the angular calculations explicitly in ∆ and
leave only the radial integral to be calculated numerically. Then
D
Z
1
1
dX1 dX2 dX3 dX4 log(X12 + X22 ) log(X32 + X42 )
4λ2 (2π)2 [ 41 − ∆2 ]
h
i
1
2
2
2
2
X + X2 + X3 + X4 − 4(X1 X3 + X2 X4 )∆ . (2.74)
× exp − 1
4[ 4 − ∆2 ] 1
E
δt(~x)δt(~0) =
Changing to polar coordinates
X1 = r1 cos θ1 , X2 = r1 sin θ1 ,
X3 = r2 cos θ2 , X4 = r2 sin θ2 ,
(2.75)
the integral becomes
D
E
δt(~x)δt(~0) =
Z 2π Z ∞
Z ∞
2
dθ
r1 dr1
r2 dr2 log(r1 ) log(r2 )
πλ2 (1 − 4∆2 ) 0
0
0
2
r1 + r22 − 4∆r1 r2 cos θ
× exp −
,
(2.76)
1 − 4∆2
62
where we redefined the angular variables as θ = θ1 − θ2 and θ̃ = θ1 + θ2 and integrated
over θ̃. Changing also the radial variables
r1 = r cos φ , r2 = r sin φ ,
D
δt(~x)δt(~0)
E
(2.77)
Z 2π Z π
Z ∞
2
1
=
dθ
dr r3 log(r cos φ) log(r sin φ)
dφ sin 2φ
2
2
πλ (1 − 4∆ ) 0
0
0
2
(1 − 2∆ sin 2φ cos θ) r
× exp −
.
(2.78)
1 − 4∆2
The radial integration can be performed analytically
Z
∞
2
dr r3 log(ar) log(br)e−cr =
(2.79)
0
1
π2
− 2 log(ab)(γ − 1 + log(c)) + 4 log(a) log(b) + log(c)(2γ − 2 + log(c)) ,
(γ − 2)γ +
8c2
6
where a = cos φ, b = sin φ, c =
1
(1−2∆
(1−4∆2 )
sin 2φ cos θ) and γ is the Euler constant
γ ≈ 0.57721.
Finally, the spectrum of the time delay field is defined by
δτ (~k) =
"
k
2π
3 Z
i~k·~
x
d3 x e
D
E
δt(~x)δt(~0)
#1/2
.
(2.80)
Calculation in the two limiting cases x → 0 and x → ∞ (or x → b in our case)
can be done analytically.
1. For x → 0 several terms in the integral diverge, since ∆ → 21 . In this case we
have only two degrees of freedom instead of four, since we consider a complex
scalar field at one point in space. The integral becomes
D
Z
dX1 dX2 −(X12 +X22 ) 2 2
1
e
log (X1 + X22 ) =
2
4λ
π
Z 2π Z ∞
1
1
π2
2 2
−r2
2
=
dθ
dr re log (r ) = 2 γ +
. (2.81)
4πλ2 0
4λ
6
0
E
~
~
δt(0)δt(0) =
63
2. The x → ∞ limit is much easier to handle. We recognize that ∆(~x) is simply the
Fourier transform of |uk |2 . Since uk is smooth, ∆(x → ∞) → 0, and therefore
δt(∞) is uncorrelated with δt(~0). Eq. (2.74) can be seen to factorize, giving
D
E D
E2
~
~
δt(∞)δt(0) = δt(0) , where
D
δt(~0)
E
Z
−1
=
dX1 dX2 log(X12 + X22 ) exp −(X12 + X22 )
2πλ
Z 2π Z ∞
1
γ
2
= −
.
dθ
r dr log re−r =
πλ 0
2λ
0
(2.82)
Combining these results,
the probability distribution for δt(~0) has a
rD we seeE that
D
E2
√
δt(~0)2 − δt(~0) = π/(2 6λ). While the first limit
standard deviation σ =
above is needed for programming the numerical calculations, since the integral of
Eq. (2.74) cannot be numerically evaluated at ~x = ~0, the second limit can be used as
a numerical check.
The same method can be applied to the exact calculation of any higher order correlation functions. Especially the non-Gaussian part of the power spectrum fN L can
be read off from the momentum space Fourier transform of the three-point correlation function in position space hδt(~x1 )δt(~x2 )δt(x3 )i = hδt(~x1 )δt(~x2 )δt(0)i. Taking the
D
E
Fourier transform we can compute δt(~k1 )δt(~k2 )δt(k3 ) , from which we can extract
the properties of the bispectrum.
The form of the three point function in position space is
−(2π)2
hδt(~x1 )δt(~x2 )δt(0)i =
λ3
Z
2π
Z
dγ1 dγ2
0
π
Z
sin θdθ
0
2π
dφF (γ1 , γ2 , θ, φ) , (2.83)
0
where F (γ1 , γ2 , θ, φ) is a function of four angular variables. Calculations regarding
the form of the bispectrum are given in the next chapter of the present thesis.
2.6
Numerical Results and Discussion
Let us begin by plotting one example of the free field theory (FFT) calculation of the
time delay power spectrum, from Eqs. (2.78) and (2.80), along with the corresponding
64
Figure 2-5: Comparison between the RSG and FFT methods. The end time was
taken to be 15 e-folds after the waterfall transition and µφ = 20 and µψ = 1/20. We
can see that the spectrum of the time delay field calculated in the free field theory
agrees very well with the rescaled version of the RSG approximation A δτRSG (Bk).
curve derived using the RSG approximation, Eq. (2.56). We use the sample parameters µψ =
1
20
and µφ = 20. Both calculations give a spike, but a spike of different
width, different height and different position. Let us rescale the RSG result as follows
δτRSG,rescaled (k) = A δτRSG (Bk) ,
(2.84)
where A and B are O(1) constants calculated by requiring the peaks of the FFT and
RSG distributions to match in position and amplitude. The results are plotted in
Fig. 2-5. We can see that the FFT and RSG curves do not seem similar. However the
rescaled RSG curve seems to follow the FFT curve very well, as was first noticed by
Burgess [13]. Based on our simulations the curves generally tend to agree better for
low wavenumbers, up to and including the peak, and start deviating after the peak.
The rescaling parameters vary with the field masses chosen and for the particular
choice of Fig. 2-5 were calculated to be A = 0.6152 and B = 3.25 . We do not yet fully
understand this behavior, but we are studying it both analytically and numerically
and will present our findings in a subsequent paper.
65
We will now do an extensive scan of parameter space {µφ , µψ } in order to have
reliable estimates on the magnitude and wavelength of the perturbations. This is
important both to make sure that CMB constraints can be satisfied as well as to
study the formation of primordial black holes that might lead to the supermassive
black holes found in the centers of galaxies. Since the original motivation for this work
has been the supersymmetric models first presented in [5] and [6], we will present the
results for the perturbations in these models. However, our quadratic approximation
holds for more general hybrid inflation models. Hence it is important to make a modelindependent parameter sweep. This will provide a more general set of predictions of
this class of models. We will give both exact power spectra, as well as try to isolate
the dominant features and provide a qualitative understanding of their dependence
on the model’s parameters.
2.6.1
Model-Independent Parameter Sweep
There are several model-dependent parameters that give us some control over the
properties of the resulting power spectrum. Initially we will fix the value of the field
at the end of inflation to be |φend | = 1014 in units of the Hubble parameter. With
this assumption (which will be relaxed later), we can calculate the properties of the
power spectrum as a function of the masses. Initially we fix the reduced timer field
mass to be µψ =
1
20
and vary the mass of the waterfall field. The results are shown
in Fig. 2-6. We have plotted (clockwise from the top left)
1. The end time of inflation, defined as the time when the RMS value of the field
reaches the end value.
2. The maximum amplitude of the spectrum of the time delay.
3. The comoving wavenumber at which the aforementioned maximum value occurs.
Thinking about black holes, this is the scale at which black holes will be most
likely produced.
4. The width of the time delay distribution in the logarithmic scale, taken as
66
Figure 2-6: Parameter sweep for constant timer field mass µψ = 1/20 and constant
end field value φend = 1014 . Data points are plotted along with a least square power
law fit. The same trend is evident in all curves. The time delay spectrum grows
in amplitude and width and is shifted towards larger momentum values as the mass
product decreases. Also inflation takes longer to end for low mass product.
67
Figure 2-7: Time delay spectra for different values of the mass product, keeping the
1
timer field mass fixed at µψ = 20
∆k = log10
k+1/2
k−1/2
where k±1/2 are the wavenumbers at which the distribution
reaches one half of its maximum value.
We see that all the plotted quantities show a decreasing behavior as one increases the
mass product. In order to quantify this statement, we fitted each set of data points
with a power law curve of the form y = axb +c. The scaling exponent b for the various
quantities was btend ≈ −0.88, bδτmax ≈ −0.34, bkmax ≈ −3.219, b∆k ≈ −1.17. As a
comparison, the corresponding best fit exponent of the growth rate λ as a function
of the mass product is bλ ≈ −0.85.
In order to get a better understanding of what these parameters actually mean,
we plot three characteristic spectra for three values of the mass ratio in Fig. 2-7.
We also rescale the spectra by the growth factor λ. This probes the actual form of
the two point correlation function, as seen in momentum space. That is, it shows
the evaluation of the spectrum in Eq. (2.80), while ignoring the factor of 1/λ2 in the
D
E
~
evaluation of δt(~x)δt(0) from Eq. (2.78).
Before continuing to a more thorough examination of parameter space, let us
understand how changing the field value at the end of inflation will change our results.
Fixing the product of the reduced masses equal to 2 (µψ =
1
20
and µφ = 40), we let
the field value φend vary by four orders of magnitude. The results are shown in
Fig. 2-8. It is evident that the curves for δτ (k) are of identical form and slightly
68
Figure 2-8: Perturbation spectrum for varying field value at tend for constant masses.
The time delay curves are identical in shape and differ only in amplitude. This
variation is entirely due to the different value of the time dependent growth factor λ,
which differs for each case because inflation simply takes longer to end for larger end
field values.
69
different magnitude. The last graph shows the product λ · δτ (k) for the two curves
at φend = 1012 and φend = 1016 , plotted respectively as a green thick and a black thin
line. It is seen, that once rescaled the two curves fall exactly on top of each other,
meaning that the actual integral that gives us the two point function in position space
is time independent, once we enter the region where all modes behave identically.
Furthermore if one takes the product of the maximum value of the time delay times
the growth parameter (δτmax ·λ) for the different values of φend the result is constant for
the range explored here to better than 1 part in 106 , meaning that they are identical
within the margins of numerical error. Thus, changing the value of the field at which
inflation ends can affect the resulting perturbation spectrum only by changing the
growth parameter λ, for which we have a very accurate analytical estimate in the
form of Eq. (2.43). From this point onward, we will keep the end field value fixed at
φend = 1014 and keep in mind that the fluctuation magnitude can change by 10% or
so if this field value changes.
Once we fix the field magnitude at the end of inflation we have two more parameters to vary, namely the two masses: the actual timer field mass and the asymptotic
tachyonic waterfall field mass. The two masses can be varied either independently on
a two dimensional plane or along some line on the plane, in a specific one-dimensional
way. Fixing one of the two masses is such a way of dimensional reduction of the available parameter space, as we did before. Another way to eliminate one of the variables
is to fix the mass product and change the mass ratio. This will prove and quantify
the statement, that (at least for heavy waterfall and light timer fields) the result is
controlled primarily by the mass product.
We fix the mass product at µφ µψ = 2. The results are shown in Fig. 2-9. The
curves are of identical form and everything is again controlled only by λ. On the top
left figure we plotted λ δτ for the two extreme values and the curves fall identically on
top of each other (color-coding is as before). Furthermore if we calculate the product
λ · δτmax for different values of the mass ratio we get a constant result 0.1225 ± 2 · 10−5
where the discrepancy can be attributed to our finite numerical accuracy. The second
70
feature of this calculation is the extremely flat part of the end-time, growth factor
and maximum time delay curves for large values of the mass ratio and the abrupt
change as the mass ratio gets smaller. For the value of the mass product that we have
chosen, this transition happens as the timer field mass approaches unity. Let us look
at the expansion of the effective waterfall field mass
2
µ2φ,ef f = µ2φ 1 − e−µ̃ψ N = µ2φ µ̃2ψ N (1 − µ̃2ψ N + ...) .
(2.85)
When the second term in the expansion cannot be neglected, the dynamics of the
problem stops being defined by the mass product alone. This explains the abrupt
change we see as we lower the mass ratio. By doing the same simulation for different
values of the fixed mass product we get similar results.
We can now do the opposite, that is fix the ratio and change the mass product.
The results are shown as the open circles in the top two diagrams of Fig. 2-10, and
in the lower diagrams of the figure. There are two main comments to be made. First
of all, in the case of a fixed ratio, the growth rate λ does not solely determine the
results. Rescaling the spectrum by λ not only fails to give a constant peak amplitude
(Fig. 2-10, lower left), but the result are spectra of different shapes (Fig. 2-10, lower
right). On the other hand, the data points taken with a constant mass ratio and a
constant timer field mass (µψ =
1
),
20
as shown by the +’s on the upper diagrams of
Fig. 2-10, fall precisely on the same curve! This clearly demonstrates that the only
relevant parameter, at least for a light timer field, is the mass product!
We see that contrary to the fixed product case, the results for fixed mass ratio do
not depend solely on λ. We have established that the the most important factor in
determining the time delay field is the product of the waterfall and timer field masses,
especially for a light timer field.
2.6.2
Supernatural Inflation
We now turn our attention to the supernatural inflation models that were studied in
[5] and [6]. We will examine each of the four cases separately.
71
Figure 2-9: Fixing the mass product at 2 and varying the mass ratio. There is
significant variation only for low mass ratio, when the light timer field approximation
loses its validity. Furthermore the curves of maximum time delay amplitude and
1/λ follow each other exactly up to our numerical accuracy. Finally by rescaling the
spectra by the growth factor λ they become identical for all values of the mass ratio.
72
Figure 2-10: Fixing the mass ratio at 900 (open circles) or the timer mass at µψ = 1/20
(+’s). The time delay spectra for different mass products show no common shape
characteristics and remain different even when rescaled by λ. Furthermore the end
time and maximum perturbation amplitude curves are identical for constant mass
ratio and constant timer field mass, proving that indeed the mass product is the
dominant parameter.
73
Figure 2-11: First Supernatural inflation model with M 0 at the Planck scale. The
spectra corresponding to the maximum and minimum mass product are shown. We
observe good agreement with the results of the model independent parameter sweep
of the previous section, because the timer field mass is much smaller than the Hubble
scale.
Let us start with the first SUSY model (described by Eq. (2.45)) with the interactionsuppressing mass scale M 0 set at the Planck scale. The mass of the timer field was
calculated to be 50 to 100 times less than the Hubble scale, while the asymptotic
waterfall field mass was more than 20 times the Hubble scale. This means that the
model is well into the region where the two masses are separated by a few orders of
magnitude. According to the analysis of the previous section, we expect the mass
product to be the dominant factor in the generation of density perturbations. In the
left part of Fig. 2-11 we see the mass product for this model. We can see that the
mass product varies less than 15%. It is hence enough to calculate the time delay
spectra for the two extreme values and say that all other values of the mass product
will fall between the two, as shown in 2-11.
Putting the mass scale M 0 of the first SUSY model at the GUT scale changes the
masses as well as the Hubble scale by one order of magnitude. However the reduced
masses and their product have very similar values as before. This is shown in Fig.
2-12
It is worth noting that these two SUSY models contain a very light timer field,
hence the results should be the same as our previous parameter space sweep with a
74
Figure 2-12: First Supernatural inflation model with M 0 at the GUT scale. The
spectra corresponding to the maximum and minimum mass product are shown. There
is again good agreement with the results of the previous section.
constant light timer field. If one compares Fig. 2-11 and Fig. 2-12 with Fig. 2-6,
we indeed see excellent agreement for the amplitude and width of the time delay
spectrum.
When setting the mass scale M 0 at some lower scale of 1011 GeV, the reduced timer
and waterfall masses become O(1). This means that in this case the parameter λ saturates faster and the perturbation spectrum reaches its asymptotic limit earlier and becomes time-independent from that point onward. Furthermore the actual value of the
growth parameter λ is smaller, leading to an enhanced perturbation amplitude, the
largest among the models studied here. The mass product changes by a factor of 2.5
as seen in Fig. 2-13. We choose five points in the allowed interval of mass values and
calculate the corresponding curves. The specific values of the mass parameter M are
M = 1.06 · 1010 GeV, 2.4 · 1010 GeV, 5.42 · 1010 GeV, 1.23 · 1011 GeV, 2.77 · 1011 GeV.
The corresponding pairs of reduced waterfall and timer masses are {µφ , 1/µψ } =
{3.19, 4.48}, {2.58, 2.97}, {2.22, 2.05}, {1.99, 1.47}, {1.83, 1.09}. The points on the
mass product graph are color coded to match the corresponding time delay curve
in Fig. 2-13. We can see that since the mass products have a larger variation, the
resulting spectra have quite different time delay spectra. Also, since the timer is not
much lighter than the Hubble scale, the curves do not scale according to our previous
analysis.
75
Figure 2-13: First Supernatural inflation model with M 0 at the intermediate scale.
Five representative pairs of masses were chosen and the corresponding time delay
curves are shown. This model can give maximum time delay of more than 0.1.
Figure 2-14: Second Supernatural inflation model. Three representative pairs of
masses were chosen and the corresponding time delay curves are shown.
We finally consider SUSY model 2, Eq. (2.46), with the ψ 2 φ2 interaction term.
Again the reduced masses are O(1), so we expect a small λ leading to a large amplitude perturbation spectrum. The mass product varies around 1 by less than ±15%.
We choose three values of the mass product (the two extrema and an intermediate one) and plot the resulting curves in Fig. 2-14. The specific values of the
mass parameter M are M = 1.080 · 1010 GeV, 1.006 · 1011 GeV, 9.376 · 1011 GeV
and the corresponding pairs of reduced waterfall and timer masses are {µφ , 1/µψ } =
{2.697, 2.367}, {2.330, 2.262}, {2.007, 2.170}.
76
2.7
Conclusions
We presented a novel method for calculating the power spectrum of density fluctuations in hybrid inflation, one that does not suffer from the non-existence of a classical
field trajectory. We used this method to numerically calculate the power spectrum
for a wide range of parameters and concluded that in the case of a light timer field,
all characteristics of the power spectrum are controlled by the product of the masses
of the two fields. In particular the amplitude was fitted to a power law and found to
behave as δτmax ∼ 0.03(µφ µψ )−0.34 and the width in log-space as ∆k ∼ 1.7(µφ µψ )−1.17 .
Furthermore we made connection to SUSY inspired models of hybrid inflation and
gave numerical results to their power spectra as well. For the SUSY models with
a light timer field the numerical results were in excellent agreement with our fitted
parameters.
Work is currently under way in refining and extending the formalism. Understanding the rescaling properties between the RSG approximation and the exact result could provide further insight into the physics of the problem and provide quasianalytical approximation of well controlled accuracy. We will also apply our results
to estimating the number and size of primordial black holes and try to make contact with astrophysical observations regarding supermassive black holes in galactic
centers. Finally we are examining the predictions of our model for the non-Gaussian
part of the perturbation spectrum.
2.8
Acknowledgements
We thank Kristin Burgess and Nguyen Thanh Son, whose earlier work on this subject paved the way for the current project. We also thank Larry Guth, who helped
us understand that this problem does not require a Monte Carlo calculation, and
Mark Hertzberg, who helped us understand the dynamics of the waterfall transition. We also thank Alexis Giguere, Illan Halpern, and Matthew Joss for helpful
discussions. The work was supported in part by the DOE under Contract No. DE77
FG02-05ER41360.
2.9
2.9.1
Appendix
Zero mode at early times
The ~k = 0 mode is not captured by the procedure described in the main text. If
we consider this mode alone for asymptotically early times, so that we keep only the
exponential in the mass term
2
ü + 3u̇ = −µ2φ e−µ̃ψ N u .
(2.86)
This can again be solved in terms of Bessel functions by defining a new variable and
a new function as
2
z̃ = αe−µ̃ψ N/2 , u(0, N ) = z̃ β Z̃(z̃) .
(2.87)
The mode function becomes
dZ̃
d2 Z̃
z̃ 2 2 + z̃
dz̃
dz̃
6
1 + 2β − 2
µ̃ψ
!
+ Z̃
6β µ2φ z̃ 2 4
β2 − 2 + 2 4
µ̃ψ
α µ̃ψ
!
=0.
(2.88)
The standard form of the differential equation that gives Bessel functions is
z
2
dZν
Zν
+
z
+ (z 2 − ν 2 )Zν = 0 .
2
dz
dz
2d
(2.89)
By appropriately choosing the constants α,β and ν the two equations can be made
identical. The choices are
β=
2µφ
3
3
, α= 2 , ν= 2 .
2
µ̃ψ
µ̃ψ
µ̃ψ
78
(2.90)
Finally introducing an arbitrary constant of normalization N0 , the solution for the
zero mode at asymptotically early times becomes
u(0, N ) = N0 e−3N/2 Hν(1) (z̃) , z̃ =
2µφ −µ̃2ψ N/2
e
.
µ̃2ψ
(2.91)
The normalization factor N0 can be defined using the Wronskian at early times. The
Wronskian at all times is defined as
∂u∗ (−~k, t) ∂u(~k, t) ∗ ~
W (~k, t) = u(~k, t)
−
u (−k, t) .
∂t
∂t
(2.92)
Taking the time derivative and using the equation of motion
∂W (~k, t)
∂ 2 u∗ (−~k, t) ∂ 2 u(~k, t) ∗ ~
= u(~k, t)
−
u (−k, t) = −3HW (~k, t)
∂t
∂t2
∂t2
⇒ W (~k, t) = f (~k)e−3Ht .
(2.93)
Since f (~k) is by definition independent of time, we will compute it at approximately
early times, where we know the solution in analytic form and the solution is
W (~k 6= 0) = ie−3N , W (~k = 0) =
2irµ2ψ H 2 −3N
N0 e
.
pi
(2.94)
Requiring that the Wronskian be a continuous function of ~k at all times we can extract
the value of N0 .
s
N0 =
3π
.
2rµ2ψ H
79
(2.95)
2.9.2
Initial Conditions
We can rewrite the mode equation as a system of three coupled first order differential
equations.
dθ
dN
dR
dN
dṘ
dN
k̃e−3N
= −
,
R2
(2.96)
= Ṙ ,
(2.97)
k̃ 2 e−6N
2
=
− 3Ṙ − e−2N k̃ 2 R + µ2φ (1 − e−µ̃ψ N )R .
3
R
(2.98)
In this notation, Ṙ is one of the three independent functions.
This is not a system of three coupled ODE’s in the strict sense. We can first solve
the two equations
dR
dN
and
dṘ
dN
as they do not contain any terms involving θ or its
derivative. We can then integrate
dθ
dN
forward in time, using the calculated values
of R(N ). Furthermore it is clear that the equations needed to calculate the spectral
quantities of the time delay field only depend on the magnitude of the wavenumber,
as was expected due to the isotropy of the problem, so we need only solve the mode
equations for one positive semi-axis.
We know from the analytical solution at early times that
R(N → −∞) → e−N .
(2.99)
Since we have to start the numerical integration at some finite negative time without
losing much in terms of accuracy, we refine the initial condition by including extra
terms in the above expression. We will then start numerically integrating when
our expansion violates the desired accuracy bound. We define the correction to the
asymptotic behavior as δR(N ) such that
R(N ) ≡ e−N + δR(N ) .
80
(2.100)
We will plug this into Eq. (2.34) and expand δR in powers of µ2φ and eN .
N h
i
i
eN
e3N
e5N
e
e3N h
2
2
2
=
−
+
+ µφ
1 − e−µ̃ψ N +
4 + e−µ̃ψ N (µ̃4ψ − 6µ̃2ψ − 4)
2k̃ 2 8k̃ 4 16k̃ 6
4k̃ 2
16k̃ 4
i
5N h
−µ̃2ψ N
8
6
4
2
2 e
−µφ
86 + e
(µ̃ψ − 14µ̃ψ + 53µ̃ψ − 25µ̃ψ − 86)
64k̃ 6
3N h
i2
e
e5N h
2
2
4
+5µφ
1 − e−µ̃ψ N −
29 − e−µ̃ψ N (9µ̃4ψ − 65µ̃ψ 2 + 58)
32k̃ 4
64k̃ 6
i
i3
5N h
−2µ̃2ψ N
−µ̃2ψ N
2
6 e
4
+e
(14µ̃ψ − 65µ̃ψ + 29) + 15µφ
1−e
+ O e7N , µ8φ . (2.101)
128k̃ 6
δR
We can now use the initial value expansion for R(N ) to calculate the corresponding
expansion for the phase θ(N ) through Eq. (2.32).
The asymptotic behavior, given by the standard definition of the Hankel functions
is
θ(N → −∞) = k̃e−N − π ⇔ θ(N → −∞) + π = k̃e−N .
(2.102)
We define θ̃ ≡ θ(N ) + π ⇒ θ̃˙ = θ̇, in order to keep track of the constant phase factor
without carrying it through the perturbation expansion.
Defining the corrections to the early time behavior of the phase as
θ̃ = e−N [k̃ + δθ(N )] ,
(2.103)
we can construct a similar expansion as the one for δR. Since our formalism does not
require knowledge of θ(N ) we will not report the full expansion. It can be calculated
in a straightforward way following the definition of Eq. (2.103), the evolution of Eq.
(2.32) and the expansion for δR given in Eq. (2.101).
81
Figure 2-15: Mode functions for µφ = 22 and m̃uψ = 1/18. The left column is
calculated for k̃ = 1/256 and the right for k̃ = 256
82
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85
86
Chapter 3
Hybrid Inflation with N Waterfall
Fields: Density Perturbations and
Constraints
Hybrid inflation models are especially interesting as they lead to a spike in the density
power spectrum on small scales, compared to the CMB. They also satisfy current
bounds on tensor modes. Here we study hybrid inflation with N waterfall fields
sharing a global SO(N ) symmetry. The inclusion of many waterfall fields has the
obvious advantage of avoiding topologically stable defects for N > 3. We find that it
also has another advantage: it is easier to engineer models that can simultaneously
(i) be compatible with constraints on the primordial spectral index, which tends to
otherwise disfavor hybrid models, and (ii) produce a spike on astrophysically large
length scales. The latter may have significant consequences, possibly seeding black
hole formation. We calculate correlation functions of the time-delay, a measure of
density perturbations, produced by the waterfall fields, as a convergent power series
in both 1/N and the field’s correlation function ∆(x). We show that for large N ,
the two-point function is hδt(x) δt(0)i ∝ ∆2 (|x|)/N and the three-point function is
hδt(x) δt(y) δt(0)i ∝ ∆(|x − y|)∆(|x|)∆(|y|)/N 2 . In accordance with the central
limit theorem, the density perturbations on the scale of the spike are approximately
Gaussian for large N and non-Gaussian for small N .
87
3.1
Introduction
Inflation, a phase of accelerated expansion in the very early universe thought to be
driven by one or several scalar fields, is our paradigm of early universe cosmology
[1, 2, 3, 4]. It naturally explains the large scale homogeneity, isotropy, and flatness
of the universe. Moreover, its basic predictions of approximate scale invariance and
small non-Gaussianity in the ∼ 10−5 level departures from homogeneity and isotropy
are in excellent agreement with recent CMB data [5, 6] and large scale structure.
While the basic paradigm of inflation is in excellent shape, no single model stands
clearly preferred. Instead the literature abounds with various models motivated by
different considerations, such as string moduli, supergravity, branes, ghosts, Standard
Model, etc [7, 8, 9, 10, 11, 12, 13, 14]. While the incoming data is at such an impressive
level that it can discriminate between various models and rule out many, such as
models that overpredict non-Gaussianity, it is not clear if the data will ever reveal
one model alone. An important way to make progress is to disfavor models based on
theoretical grounds (such as issues of unitarity violation, acausality, etc) and to find
a model that is able to account for phenomena in the universe lacking an alternate
explanation. It is conceivable that some version of the so-called “hybrid inflation”
model may account for astrophysical phenomena, for reasons we shall come to.
The hybrid inflation model, originally proposed by Linde [15], requires at least two
fields. One of the fields is light and another of the fields is heavy (in Hubble units).
The light field, called the “timer”, is at early times slowly rolling down a potential
hill and generates the almost scale invariant spectrum of fluctuations observed in the
CMB and in large scale structure. The heavy field, called the “waterfall” field, has
an effective mass that is time-dependent and controlled by the value of the timer
field. The waterfall field is originally trapped at a minimum of its potential, but as
its effective mass-squared becomes negative, a tachyonic instability follows, leading to
the end of inflation; an illustration is given in Figure 3-1. The name “hybrid inflation”
comes from the fact that this model is a sort of hybrid between a chaotic inflation
model and a symmetry breaking inflation model.
88
As originally discussed in Refs. [16, 17], one of the most fascinating features of
hybrid models is that the tachyonic behavior of the waterfall field leads to a sharp
“spike” in the density power spectrum. This could seed primordial black holes [18,
19, 20, 21, 22]. For generic parameters, the length scale associated with this spike
is typically very small. However, if one could find a parameter regime where the
waterfall phase were to be prolonged, lasting for many e-foldings, say Nw ∼ 30 − 40,
then this would lead to a spike in the density perturbations on astrophysically large
scales (but smaller than CMB scales). This may help to account for phenomena
such as supermassive black holes or dark matter, etc. Of course the details of all
this requires a very careful examination of the spectrum of density perturbations,
including observational constraints.
Ordinarily the spectrum of density perturbations in a given model of inflation
is obtained by decomposing the inflaton field into a homogeneous part plus a small
inhomogeneous perturbation. However, for the waterfall fields of hybrid inflation,
this approach fails since classically the waterfall field would stay forever at the top
of a ridge in its potential. It is the quantum perturbations themselves that lead to a
non-trivial evolution of the waterfall field, and therefore the quantum perturbations
cannot be treated as small. Several approximations have been used to deal with this
problem [16, 17, 25, 26, 27, 28, 29, 30, 31, 23, 24]. Here we follow the approach
presented by some of us recently in Ref. [32], where a free field time-delay method
was used, providing accurate numerical results.
In this chapter, we generalize the method of Ref. [32] to a model with N waterfall
fields sharing a global SO(N ) symmetry. A model of many fields may be natural in
various microscopic constructions, such as grand unified models, string models, etc.
But apart from generalizing Ref. [32] to N fields, we also go much further in our
analysis: we derive explicit analytical results for several correlation functions of the
so-called time-delay. We formulate a convergent series expansion in powers of 1/N
and the field’s correlation function ∆(x). We find all terms in the series to obtain the
two-point correlation function of the time-delay for any N . We also obtain the leading
order behavior at large N for the three-point function time-delay, which provides a
89
measure of non-Gaussianity. We find that the non-Gaussianity is appreciable for small
N and suppressed for large N .
We also analyze in detail constraints on hybrid inflation models. We comment
on how multiple fields avoids topological defects, which is a serious problem for low
N models. However, the most severe constraint on hybrid models comes from the
requirement to obtain the observed spectral index ns . We show that at large N ,
it is easier to engineer models that can fit the observed ns , while also allowing for a
prolonged waterfall phase. This means that large N models provide the most plausible
way for the spike to appear on astrophysically large scales and be compatible with
other constraints.
The organization of this chapter is as follows: In Section 3.2 we present our hybrid
inflation model and discuss our approximations. In Section 3.3 we present the timedelay formalism, adapting the method of Ref. [32] to N fields. In section 3.4 we derive
a series expansion for the two-point function, we derive the leading order behavior of
the three-point function, and we derive results in k-space. In Section 3.5 we present
constraints on hybrid models, emphasizing the role that N plays. In Section 3.6 we
discuss and conclude. Finally, in the Appendices we present further analytical results.
3.2
N Field Model
The model consists of two types of fields: The timer field ψ that drives the first
slow-roll inflation phase, and the waterfall field φ that becomes tachyonic during the
second phase causing inflation to end. In many hybrid models, φ is comprised of two
components, a complex field, but here we allow for N real components φi . We assume
the components share a global SO(N ) symmetry, and so it is convenient to organize
them into a vector
~ = {φ1 , φ2 , . . . , φN }.
φ
(3.1)
For the special case N = 2, this can be organized into a complex field by writing
√
φcomplex = (φ1 + i φ2 )/ 2.
The dynamics is governed by the standard two derivative action for the scalar
90
~ minimally coupled to gravity as follows (signature + − −−)
fields ψ, φ
Z
S=
√
"
d4 x −g
1
1 µν
R +
g ∂µ ψ∂ν ψ
16πG
2
#
1 µν ~
~ − V (ψ, φ) .
+ g ∂µ φ · ∂ν φ
2
(3.2)
The potential V is given by a sum of terms: V0 providing false vacuum energy, Vψ (ψ)
governing the timer field, Vφ (φ) governing the waterfall field, and Vint (ψ, φ) governing
their mutual interaction, i.e.,
V (ψ, φ) = V0 + Vψ (ψ) + Vφ (φ) + Vint (ψ, φ).
(3.3)
During inflation we assume that the constant V0 dominates all other terms.
The timer field potential Vψ and the waterfall field potential Vφ can in general be
complicated. In general, they are allowed to be non-polynomial functions as part of
some low energy effective field theory, possibly from supergravity or other microscopic
theories. For our purposes, it is enough to assume an extremum at ψ = φ = 0 and
expand the potentials around this extremum as follows:
1
Vψ (ψ) = m2ψ ψ 2 + . . .
2
1
~·φ
~ + ...
Vφ (φ) = − m20 φ
2
(3.4)
(3.5)
The timer field is assumed to be light mψ < H and the waterfall field is assumed to be
heavy m0 > H, where H is the Hubble parameter. In the original hybrid model, this
quadratic term for Vψ was assumed to be the entire potential. This model leads to a
spectrum with a spectral index ns > 1 and is observationally ruled out. Instead we
need higher order terms, indicated by the dots, to fix this problem. This also places
constraints on the mass of the timer field mψ , which has important consequences. We
discuss these issues in detail in Section 3.5.
As indicated by the negative mass-squared, the waterfall field is tachyonic around
91
Figure 3-1: An illustration of the evolution of the effective potential for the waterfall
field φ as the timer field ψ evolves from “high” values at early times, to ψ = ψc , and
finally to “low” values at late times. In the process, the effective mass-squared of φ
evolves from positive, to zero, to negative (tachyonic).
φ = ψ = 0. This obviously cannot be the entire potential because then the potential
would be unbounded from below. Instead there must be higher order terms that
stabilize the potential with a global minimum near V ≈ 0 (effectively setting the
late-time cosmological constant).
The key to hybrid inflation is the interaction between the two fields. For simplicity,
we assume a standard dimension 4 coupling of the form
1
~ · φ.
~
Vint (ψ, φ) = g 2 ψ 2 φ
2
(3.6)
This term allows the waterfall field to be stabilized at φ = 0 at early times during
slow-roll inflation when ψ is displaced away from zero, and then becomes tachyonic
once the timer field approaches the origin; this is illustrated in Figure 3-1.
3.2.1
Approximations
We assume that the constant V0 is dominant during inflation, leading to an approximate de Sitter phase with constant Hubble parameter H. By assuming a flat FRW
92
background, the scale factor is approximated as
a = exp(Ht).
(3.7)
At early times, ψ is displaced from its origin, so φ = 0. This means that we can
approximate the ψ dynamics by ignoring the back reaction from φ. The fluctuations in
ψ establish nearly scale invariant fluctuations on large scales, which we shall return to
in Section 3.5. However, for the present purposes it is enough to treat ψ as a classical,
homogeneous field ψ(t). We make the approximation that we can neglect the higher
order terms in the potential Vψ in the transition era, leading to the equation of motion
ψ̈ + 3H ψ̇ + m2ψ ψ = 0.
(3.8)
Solving this equation for ψ(t), we insert this into the equation for φi . We allow spatial
dependence in φi , and ignore, for simplicity, the higher order terms in Vφ , leading to
the equation of motion
φ̈i + 3H φ̇i − e−2Ht ∇2 φi + m2φ (t)φi = 0.
(3.9)
Here we have identified an effective mass-squared for the waterfall field of
m2φ (t) ≡ −m20
1−
ψ(t)
ψc
2 !
,
(3.10)
where the dimension 4 coupling g has been traded for a coupling ψc as g 2 = m20 /ψc2 .
The quantity ψc has the physical interpretation as the “critical” value of ψ such that
the effective mass of the waterfall field passes through zero. So at early times for
ψ > ψc , then m2φ > 0 and φi is trapped at φi = 0, while at late times for ψ < ψc ,
then m2φ < 0 and φi is tachyonic and can grow in amplitude, depending on the mode
of interest; Figure 3-1 illustrates these features.
93
3.2.2
Mode Functions
Since we ignore the back-reaction of φ onto ψ and since we treat ψ as homogeneous
in the equation of motion for φ (eq. (3.9)), then by passing to k-space, all modes
are decoupled. Each waterfall field φi can be quantized and expanded in modes in
momentum space as follows
Z
φi (~x, t) =
i
d3 k h
†
−ik·x ∗
ik·x
e
u
(t)
,
c
e
u
(t)
+
c
k,i
k
k
k,i
(2π)3
(3.11)
where c†k (ck ) are the creation (annihilation) operators, acting on the φi = 0 vacuum.
By assuming an initial Bunch-Davies vacuum for each φi , the mode functions uk are
the same for all components due to the SO(N ) symmetry. To be precise, we assume
that at asymptotically early times the mode functions are the ordinary Minkowski
space mode functions, with the caveat that we need to insert factors of the scale factor
a to convert from physical wavenumbers to comoving wavenumbers, i.e.,
uk (t) →
e−i k t/a
√ ,
a 2k
at early times.
(3.12)
At late times the mode functions behave very differently. Since the field becomes
tachyonic, then the mode functions grow exponentially at late times. The transition
depends on the wavenumber of interest. The full details of the mode functions were
explained very carefully in Ref. [32], and the interested reader is directed to that
paper for more information.
Now since we are approximating φi as a free field, then its fluctuations are entirely
Gaussian and characterized entirely by its equal time two-point correlation function
hφi (x) φj (y)i. Passing to k-space, and using statistical isotropy and homogeneity
of the Bunch-Davies vacuum, the fluctuations are equally well characterized by the
so-called power spectrum Pφ (k), defined through
hφi (k1 ) φj (k2 )i = (2π)3 δ(k1 + k2 ) δij Pφ (k1 ).
94
(3.13)
105
Power Pftilde
10
0.001
10-7
10-11
10-15
0.01
0.1
1
10
Wavenumber k
100
1000
Figure 3-2: A plot of the (re-scaled) field’s power spectrum Pφ̃ as a function of
wavenumber k (in units of H) for different masses: blue is m0 = 2 and mψ = 1/2,
red is m0 = 4 and mψ = 1/2, green is m0 = 2 and mψ = 1/4, orange is m0 = 4 and
mψ = 1/4.
This means the power spectrum is
Z
δij Pφ (k) =
d3 x e−ik·x hφi (x)φj (0)i.
(3.14)
It is simple to show that the power spectrum is related to the mode functions by
Pφ (k) = |uk |2 .
(3.15)
In Figure 3-2 we plot a rescaled version of Pφ , where we divide out by the root-meansquare (rms) of φ as defined in the next Section.
3.3
The Time-Delay
We would now like to relate the fluctuations in the waterfall field φ to a fluctuation
in a physical observable, namely the density perturbation. An important step in this
direction is to compute the so-called “time-delay” δt(x); the time offset for the end of
inflation for different parts of the universe. This causes different regions of the universe
to have inflated more than others, creating a difference in their densities (though we
will not explicitly compute δρ/ρ here). This basic formalism was first introduced by
95
Hawking [33] and by Guth and Pi [34], and has recently been reviewed in Ref. [35].
In the context of hybrid inflation, it was recently used by some of us in Ref. [?]. It
provides an intuitive and straightforward way to calculate primordial perturbations
and we now use this to study perturbations established by the N waterfall fields.
In its original formulation, the time-delay formalism starts by considering a classical homogeneous trajectory φ0 = φ0 (t), and then considers a first order perturbation
around this. At first order, one is able to prove that the fluctuating inhomogeneous
field φ(x, t) is related to the classical field φ0 , up to an overall time offset δt(x),
φ(x, t) = φ0 (t − δt(x)).
(3.16)
In the present case of hybrid inflation, the waterfall field is initially trapped at
φ = 0 and then once it becomes tachyonic, it eventually falls off the hill-top due to
quantum fluctuations. This means that there is no classical trajectory about which
to expand. Nevertheless, we will show that, to a good approximation, the field φ(x, t)
is well described by an equation of the form (3.16), for a suitably defined φ0 . The key
is that all modes of interest grow at the same rate at late times. Further information
of the time evolution of the mode functions is in Ref. [32].
To show this, we need to compute the evolution of the field φ according to eq. (3.9).
This requires knowing mφ (t), which in turn requires knowing ψ(t). By solving eq. (3.8)
for the timer field and dispensing with transient behavior, we have
ψ(t) = ψc exp (− p t) ,
where

s
3
p=H −
2

2
m
9
ψ
− 2,
4 H
(3.17)
(3.18)
(note p > 0). We have set the origin of time t = 0 to be when ψ = ψc and assume
ψ > ψc at early times.
Substituting this solution into mφ (t), we can, in principle, solve eq. (3.9). In
general, the solution is somewhat complicated with a non-trivial dependence on
96
wavenumber. However, at late times the behavior simplifies. Our modes of interest are super-horizon at late times. For these modes, the gradient term is negligible
and the equation of motion reduces to
φ̈i + 3H φ̇i + m2φ (t)φi = 0.
(3.19)
So each mode evolves in the same way at late times. Treating mφ (t) as slowly varying
(which is justified because the timer field mass mψ < H and so p is small), we can
solve for φi at late times t in the adiabatic approximation. We obtain
Z
t
φi (x, t) = φi (x, t0 ) exp
0
0
dt λ(t ) ,
(3.20)
t0
where
λ(t) = H
3
− +
2
r
!
9 m20
+
1 − e−2 p t .
4 H2
(3.21)
Here t0 is some reference time. For t > 0, we have λ(t) > 0, so the modes grow
exponentially in time. Later in Section 3.5 we explain that in fact λ is roughly
constant in the latter stage of the waterfall phase, i.e., the exp(−2 p t) piece becomes
small.
We now discuss fluctuations in the time at which inflation ends. For convenience,
we define the reference time t0 to be the time at which the rms value of the field
reaches φend ; the end of inflation
N φ2rms (t0 ) = φ2end ,
(3.22)
where we have included a factor of N to account for all fields, allowing φrms to refer
to fluctuations in a single component φi , i.e., φ2rms = hφ2i (0)i. In terms of the power
spectrum, it is
φ2rms
Z
=
d3 k
Pφ (k).
(2π)3
(3.23)
If we were to include arbitrarily high k, this would diverge quadratically, which is
the usual Minkowski space divergence. However, our present analysis only applies to
97
modes that are in the growing regime. For these modes, Pφ (k) falls off exponentially
with k and there is no problem. So in this integral, we only include modes that are
in the asymptotic regime, or, roughly speaking, only super-horizon modes.
Using eq. (3.20), we can express the field φi (x, t) at time t = t0 + δt in terms of
the field φi (x, t0 ) by
φi (x, t) = φi (x, t0 ) exp (λ(t0 ) δt) .
(3.24)
If t is chosen to be the time tend (x) at which inflation ends at each point in space,
~ φ
~ x, tend (x) = φ2 = N φ2 (t0 ), and the above equation becomes
then φ·
rms
end
~ t0 )· φ(x,
~ t0 ) exp (2 λ(t0 ) δt),
N φ2rms (t0 ) = φ(x,
(3.25)
which can be solved for the time-delay field δt(x) = tend (x) − t0 as
~ t0 )· φ(x,
~ t0 )
−1
φ(x,
δt(x) =
ln
2λ(t0 )
N φ2rms (t0 )
!
.
(3.26)
This finalizes the N component analysis of the time-delay, generalizing the two component (complex) analysis of Ref. [32].
3.4
Correlation Functions
We now derive expressions for the two-point and three-point correlation functions of
the time-delay field. To do so, it is convenient to introduce a re-scaled version of the
correlation function ∆(x) defined through
hφi (x)φj (0)i = ∆(x)φ2rms δij .
(3.27)
By definition ∆(0) = 1, and as we vary x, ∆ covers the interval ∆(x) ∈ (0, 1].
(Ref. [32] used a different convention where ∆ covers the interval ∆(x) ∈ (0, 1/2]).
In Figure 3-3, we show a plot of ∆ at tend as a function of x, measured in Hubble
lengths (H −1 ), for a choice of masses.
98
1.0
Field Correlation D
0.8
0.6
0.4
0.2
0.0
0.01
0.1
1
Length x
10
100
Figure 3-3: A plot of the field’s correlation ∆ as a function of x (in units of H −1 ) for
different masses: blue is m0 = 2 and mψ = 1/2, red is m0 = 4 and mψ = 1/2, green
is m0 = 2 and mψ = 1/4, orange is m0 = 4 and mψ = 1/4.
3.4.1
Two-Point Function
We now express the time-delay correlation functions as a power series in ∆ and 1/N .
An alternative derivation of the power spectra of the time-delay field in terms of an
integral, which is closer to the language of Ref. [32], can be found in Appendix 3.8.3.
Using the above approximation for δt in eq. (3.26), the two-point correlation
function for the time-delay is
*
~x · φ
~x
φ
(2λ)2 hδt(x)δt(0)i = ln
N φ2rms
!
~0 · φ
~0
φ
ln
N φ2rms
!+
,
(3.28)
~ x ≡ φ(x).
~
where we have used the abbreviated notation φ
The two-point function will
include a constant (independent of x) for a non-zero hδti. This can be reabsorbed
into a shift in t0 , whose dependence we have suppressed here, and so we will ignore
the constant in the following computation. This means that we will compute the
connected part of the correlation functions.
We would like to form an expansion, but we do not have a classical trajectory about
~·φ
~ should
which to expand. Instead we use the following idea: we recognize that φ
be centralized around its mean value of N φ2rms , plus relatively small fluctuations at
99
large N . This means that it is convenient to write
~·φ
~
φ
=1+
N φ2rms
!
~·φ
~
φ
−1
N φ2rms
(3.29)
and treat the term in parenthesis on the right as small, as it represents the fluctuations
from the mean. This allows us to Taylor expand the logarithm in powers of
Φ≡
~·φ
~
φ
− 1,
N φ2rms
(3.30)
with hΦi = 0. Now recall that the series expansion of the logarithm for small Φ is
ln(1 + Φ) = Φ −
Φ2 Φ3 Φ4
+
−
+ ...,
2
3
4
(3.31)
allowing us to expand to any desired order in Φ.
The leading non-zero order is quadratic ∼ Φ2 . It is
(2λ)2 hδt(x)δt(0)i2 = hΦ(x)Φ(0)i,
hφi (x)φi (x)φj (0)φj (0)i
=
− 1,
(N φ2rms )2
(3.32)
where we are implicitly summing over indices in the second line (for simplicity, we
will place all component indices (i, j) downstairs). Using Wick’s theorem to perform
the four-point contraction, we find the result
(2λ)2 hδt(x)δt(0)i2 =
2∆2 (x)
,
N
(3.33)
where x ≡ |x|. This provides the leading approximation for large N , or small ∆.
This should be contrasted to the RSG approximation used in Refs. [16, 17] in which
the correlation function is approximated as ∼ ∆, rather than ∼ ∆2 , at leading order.
For brevity, we shall not go through the result at each higher order here, but we
report on results at higher order in Appendix 3.8.1. By summing those results to
100
different orders, we find
(2λ)2 hδt(x)δt(0)i =
2∆2 2∆4 4∆4 16∆6 8∆4
+ 2 − 3 +
+
N
N
N 3N 3 N 4
32∆6 24∆8
1
− 4 +
,
+O
4
N
N
N5
(3.34)
(where ∆ = ∆(x) here). We find that various cancellations have occurred. For
example, the −8∆2 /N 2 term that enters at cubic order (see Appendix 3.8.1) has
canceled.
Note that at a given order in 1/N there are various powers of ∆2 . However, by
collecting all terms at a given power in ∆2 , we can identify a pattern in the value of
its coefficients as functions of N . We find
(2λ)2 hδt(x)δt(0)i
2∆2 (x)
2∆4 (x)
16∆6 (x)
=
+
+
N
N (N + 2) 3N (N + 2)(N + 4)
24∆8 (x)
+ ...
+
N (N + 2)(N + 4)(N + 6)
(3.35)
We then identify the entire series as
2
(2λ) hδt(x)δt(0)i =
∞
X
Cn (N ) ∆2n (x),
(3.36)
n=1
where the coefficients are
1
Cn (N ) = 2
n
with
a
b
N
2
−1+n
n
−1
,
(3.37)
the binomial coefficient. This series is convergent for any N and ∆ ∈ (0, 1],
and is one of our central results.
For the case of a single scalar or a doublet (complex field), the full series organizes
101
itself into known functions. For N = 1 we find
2∆4 (x) 16∆6 (x)
+
+ ...
3
45
= 2 (sin−1 ∆(x))2 .
(3.38)
∆4 (x) ∆6 (x)
+
+ ...
22
32
= Li2 (∆2 (x)),
(3.39)
(2λ)2 hδt(x)δt(0)i = 2∆2 (x) +
For N = 2 we find
(2λ)2 hδt(x)δt(0)i = ∆2 (x) +
where Lis (z) is the polylogarithm function. We also find that for any even value of
N , the series is given by the polylogarithm function plus a polynomial in ∆; this is
described in Appendix 3.8.2.
Using the power series, we can easily obtain plots of the two-point function for any
N . For convenience, we plot the re-scaled quantity N (2λ)2 hδt(x)δt(0)i as a function
of ∆ in Figure 3-4 (top) for different N . We see convergence of all curves as we
increase N , which confirms that the leading behavior of the (un-scaled) two-point
function is ∼ 1/N . In Figure 3-4 (bottom) we plot N hδt(x)δt(0)i as a function of x
for different masses and two different N .
3.4.2
Three-Point Function
The three-point function is given by a simple modification of eq. (3.28), namely
−(2λ)3 hδt(x)δt(y)δt(0)i
*
!
!
!+
~y · φ
~y
~0 · φ
~0
~x · φ
~x
φ
φ
φ
ln
ln
.
= ln
N φ2rms
N φ2rms
N φ2rms
102
(3.40)
Two-Point Correlation NH2lL2 <dtHxLdtH0L>
4
3
2
1
0
0.0
0.2
0.4
0.6
Field Correlation D
0.8
1.0
Two-Point Correlation N<dtHxLdtH0L>
1.0
0.8
0.6
0.4
0.2
0.0
0.01
0.1
1
Length x
10
100
Figure 3-4: Top: a plot of the (re-scaled) two-point function of the time-delay
N (2λ)2 hδt(x)δt(0)i as a function of ∆ ∈ [0, 1] as we vary N : dot-dashed is N = 1,
solid is N = 2, dotted is N = 6, and dashed is N → ∞. Bottom: a plot of the
(re-scaled) two-point function of the time-delay N hδt(x)δt(0)i as a function of x for
different masses: blue is m0 = 2 and mψ = 1/2, red is m0 = 4 and mψ = 1/2, green
is m0 = 2 and mψ = 1/4, orange is m0 = 4 and mψ = 1/4, with solid for N = 2 and
dashed for N → ∞.
103
Here we will work only to leading non-zero order, which in this case is cubic. We
expand the logarithms as before to obtain
−(2λ)3 hδt(x)δt(y)δt(0)i3
= hΦ(x)Φ(y)Φ(0)i,
hφi (x)φi (x)φj (y)φj (y)φk (0)φk (0)i
=
+2
(N φ2rms )3
hφi (x)φi (x)φj (y)φj (y)i
−
+ 2 perms ,
(N φ2rms )2
(3.41)
where “perms” is short for permutations under interchanging x, y, 0. Using Wick’s
theorem to perform the various contractions, we find the result
−(2λ)3 hδt(x)δt(y)δt(0)i3 =
8 ∆(|x − y|) ∆(x) ∆(y)
.
N2
(3.42)
We would now like to use the three-point function as a measure of non-Gaussianity.
For a single random variable, a measure of non-Gaussianity is to compute a dimensionless ratio of the skewness to the 3/2 power of the variance. For a field theory,
we symmetrize over variables, and define the following measure of non-Gaussianity
in position space
hδt(x)δt(y)δt(0)i
S≡p
.
hδt(x)δt(y)ihδt(x)δt(0)ihδt(y)δt(0)i
(3.43)
(Recall that we are ignoring hδti, as it can be just re-absorbed into t0 , so the threepoint and two-point functions are the connected pieces). Computing this at the
leading order approximation using eqs. (3.33, 3.42) (valid for large N , or small ∆),
we find
r
S≈−
8
.
N
(3.44)
Curiously, the dependence on x, y has dropped out at this order. We see that for
small N there is significant non-Gaussianity, while for large N the theory becomes
Gaussian, as expected from the central limit theorem.
104
3.4.3
Momentum Space
Let us now present our results in k-space. We shall continue to analyze the results at
high N , or small ∆, which allows us to just include the leading order results.
For the two-point function, we define the power spectrum through
hδt(k1 )δt(k2 )i = (2π)3 δ(k1 + k2 )Pδt (k1 ).
(3.45)
We use eq. (3.33) and Fourier transform to k-space using the convolution theorem. To
do so it is convenient to introduce a dimensionless field φ̃i ≡ φi /φrms and introduce
the corresponding power spectrum Pφ̃ (k) = Pφ (k)/φ2rms = |uk |2 /φ2rms , which is the
Fourier transform of ∆(x). We then find the result
1
Pδt (k) ≈ 2
2λ N
Z
d3 k 0
P (k 0 )Pφ̄ (|k − k0 |).
(2π)3 φ̄
(3.46)
A dimensionless measure of fluctuations is the following
Pδt (k) ≡
k 3 H 2 Pδt (k)
,
2π 2
(3.47)
The factor of k 3 /(2π 2 ) is appropriate as this gives the variance per log interval:
R
h(Hδt)2 i = d ln k Pδt (k). By studying eq. (3.46), one can show Pδt ≈ const for
small k and falls off for large k. This creates a spike in Pδt (k) at a finite k ∗ and its
amplitude is rather large. (This is to be contrasted with the usual fluctuations in de
Sitter space, Pδt (k) ∝ 1/k 3 , making Pδt (k) approximately scale invariant.) We see
that the amplitude of the spike scales as ∼ 1/N , and so it is reduced for large N . A
plot of Pδt (k) is given in Figure 3-5.
For the three-point function, we define the bispectrum through
hδt(k1 )δt(k2 )δt(k3 )i = (2π)3 δ(k1 + k2 + k3 )Bδt (k1 , k2 , k3 )
(3.48)
where we have indicated that the bispectrum only depends on the magnitude of the
3 k-vectors, with the constraint that the vectors sum to zero. We use eq. (3.42) and
105
2 p2
Dimensionless Power spectrum
N k3 Pdt
0.20
0.15
0.10
0.05
0.00
0.01
0.1
1
10
Wavenumber k
100
1000
Figure 3-5: The dimensionless power spectrum N Pδt (k) at large N as a function of
k (in units of H) for different choices of masses: blue is m0 = 2 and mψ = 1/2, red
is m0 = 4 and mψ = 1/2, green is m0 = 2 and mψ = 1/4, orange is m0 = 4 and
mψ = 1/4.
Fourier transform to k-space, again using the convolution theorem. We find
1
Bδt (k1 , k2 , k3 ) ≈ − 3 2 ×
3λ N
Z 3
dk
P (k)Pφ̄ (|k1 − k|)Pφ̄ (|k2 + k|) + 2 perms
(2π)3 φ̄
(3.49)
To measure non-Gaussianity in k-space, it is conventional to introduce the dimensionless fN L parameter, defined as1
fN L (k1 , k2 , k3 ) ≡
Bδt (k1 , k2 , k3 )
.
Pδt (k1 )Pδt (k2 ) + 2 perms
(3.50)
By substituting the above expressions for Pδt and Bδt , we see that fN L is independent of N at this leading order. However, this belies the true dependence of nonGaussianity on the number of fields. This is because fN L is a quantity that can be
large even if the non-Gaussianity is relatively small (for example, on CMB scales,
any fN L smaller than O(105 ) is a small level of non-Gaussianity). Instead a more
appropriate measure of non-Gaussianity in k-space is to compute some ratio of the
bispectrum to the 3/2 power of the power spectrum, analogous to the position space
1
A factor of 6/5 is often included when studying the gauge invariant quantity ζ that appears in
cosmological perturbation theory, but it does not concern us here.
106
3.0
Dimensionless Bispectrum
N »FNL »
2.5
2.0
1.5
1.0
0.5
0.0
0.01
0.1
1
10
Wavenumber k
Figure 3-6: The dimensionless bispectrum
at large N for m0 = 4 and mψ = 1/2.
√
100
1000
N FN L as a function of k (in units of H)
definition in eq. (3.43). For the simple equilateral case, k1 = k2 = k3 , we define
k 3/2 Bδt (k)
√
FN L (k) ≡
,
3 2 πPδt (k)3/2
(3.51)
√
where we have inserted a factor of k 3/2/( 2 π) from measuring the fluctuations per
√
log interval. Using eqs. (3.46, 3.49), we see that FN L ∝ 1/ N , as we found in
position space. In Figure 3-6, we plot this function. We note that although the nonGaussianity is large, the peak is on a length scale that is small compared to the CMB
and so it evades recent bounds [6].
3.5
Constraints on Hybrid Models
Hybrid inflation models must satisfy several observational constraints. Here we discuss these constraints, including the role that N plays, and discuss the implications
for the scale of the density spike.
3.5.1
Topological Defects
The first constraint on hybrid models concerns the possible formation of topological
defects. Since the waterfall field starts at φ = 0 and then falls to some vacuum, it
107
spontaneously breaks a symmetry. For a single field N = 1, this breaks a Z2 symmetry; see Figure 3-1. For multiple fields N > 1, this breaks an SO(N ) symmetry.
N = 1 leads to the formation of domain walls, which are clearly ruled out observationally, so these models are strongly disfavored; N = 2 leads to the formation of
cosmic strings, which have not been observed and if they exist are constrained to be
small in number. This would require a very large number of e-foldings of the waterfall
phase to make compatible with observations, and seems unrealistic; N = 3 leads to
the formation of monopoles, which are somewhat less constrained; N = 4 leads to the
formation of textures, which are relatively harmless; N > 4 avoids topological defects
altogether. So choosing several waterfall fields is preferred by current constraints on
topological defects.
3.5.2
Inflationary Perturbations
Inflation generates fluctuations on large scales which are being increasingly constrained by data. An important constraint on any inflation model is the bound on
the tensor-to-scalar ratio r. Recent CMB measurements from Planck places an upper
bound on tensor modes of r < 0.11 (95% confidence) [6]. The amplitude of tensor
modes is directly set by the energy density during inflation. Typical hybrid models
are at relatively low energy scales, without the need for extreme fine tuning, and so
they immediately satisfy this bound.
Although hybrid inflation passes the tensor mode constraint with flying colors, its
ability to pass the scalar mode constraint is much less clear. The tilt of the scalar
mode spectrum is characterized by the primordial spectral index ns . WMAP [5] and
recent Planck measurements [6] place the spectral index near
ns,obs ≈ 0.96,
(3.52)
giving a red spectrum. Here we examine the constraints imposed on hybrid models
in order to obtain this value of ns .
The tilt on large scales is determined by the timer field ψ. For low scale models
108
of inflation, such as hybrid inflation, the prediction for the spectral index is
where
ns = 1 + 2 η,
(3.53)
Vψ00
1 Vψ00
η≈
=
.
8πG V0
3 H2
(3.54)
This is to be evaluated Ne e-foldings from the end of inflation, where Ne = 50 − 60 in
typical models. Combining the above equations, we need to satisfy Vψ00 ≈ −0.06 H 2 .
If we take Vψ = m2ψ ψ 2 /2, then Vψ00 > 0, and ns > 1, which is ruled out. So we need
higher order terms in the potential to cause it to become concave down at large values
of ψ where η is evaluated, while leaving the quadratic approximation for Vψ valid at
small ψ. For most reasonable potential functions, such as potentials that flatten at
large field values, we expect |Vψ00 | . m2ψ . So this suggests a bound
m2ψ & 0.06 H 2 ,
(3.55)
which can only be avoided by significant fine tuning of the potential. Hence although
the timer field is assumed light (mψ < H), it cannot be extremely light.
3.5.3
Implications for Scale of Spike
The length scale associated with the spike in the spectrum is set by the Hubble length
during inflation H −1 , red-shifted by the number of e-foldings of the waterfall phase
Nw . Since the Hubble scale during inflation is typically microscopic, we need the
duration of the waterfall phase Nw to be significant (e.g., Nw ∼ 30 − 40) to obtain a
spike on astrophysically large scales. Here we examine if this is possible.
Since we have defined t = 0 to be when the transition occurs (ψ = ψc ), then
Nw = Ht with final value at t = tend . To determine the final value, we note that modes
grow at the rate λ, derived earlier in eq. (3.21). For mψ < H, we can approximate
the parameter p (eq. (3.18)) as p ≈ m2ψ /3H. Using the above spectral index bound in
eq. (3.55), we see that for significantly large Nw (e.g., Nw ∼ 30 − 40) the exponential
109
factor
2 m2ψ Nw
exp(−2 p t) ≈ exp −
3 H2
(3.56)
is somewhat small and we will ignore it here. In this regime, the dimensionless growth
rate λ/H can be approximated as a constant
λ
3
≈− +
H
2
r
9 m20
+
.
4 H2
(3.57)
The typical starting value for φ is roughly of order H (de Sitter fluctuations) and the
typical end value for φ is roughly of order MP l (Planck scale). For self consistency,
φ must pass from its starting value to its end value in Nw e-foldings with rate set by
λ/H. This gives the approximate value for Nw as2
H
MP l
Nw ≈
ln
.
λ
H
(3.58)
This has a clear consequence: If we choose m0 H, as is done in some models of
hybrid inflation, then H/λ 1. So unless we push H to be many, many orders of
magnitude smaller then MP l , then Nw will be rather small. This will lead to a spike
in the spectrum on rather small scales and possibly ignorable to astrophysics.
Note that if we had ignored the spectral index bound that leads to eq. (3.55), then
we could have taken mψ arbitrarily small, leading to an arbitrarily small p value. In
this (unrealistic) limit, it is simple to show
2 m2ψ m20 Nw
λ
≈
.
H
9 H4
(3.59)
So by taking mψ arbitrarily small, λ could be made small, and Nw could easily be
made large. However, the existence of the spectral index bound essentially forbids
this, requiring us to go in a different direction.
The only way to increase Nw and satisfy the spectral index bound on mψ is to
2
A better approximation comes from
R t tracking the full time dependence of λ and integrating
the argument of the exponential exp( dt0 λ(t0 )), but this approximation suffices for the present
discussion.
110
take m0 somewhat close to H. This allows H/λ to be appreciable from eq. (3.57).
For instance, if we set m0 = 1.3 H, then H/λ ≈ 2. If we then take H just a few
orders of magnitude below MP l , say H ∼ 10−6,7 MP l , which is reasonable for inflation
models, we can achieve a significant value for Nw . This will lead to a spike in the
spectrum on astrophysically large scales, which is potentially quite interesting. It is
possible that there will be distortions in the spectrum by taking m0 close to H, but
we will not explore those details here. However, there is an important consequence
that we explore in the next subsection.
3.5.4
Eternal Inflation
Since we are being pushed towards a somewhat low value of m0 , near H, we need
to check if the theory still makes sense. One potential problem is that the theory
may enter a regime of eternal inflation. This could occur for the waterfall field at the
hill-top. This would wipe out information of the timer field, which established the
approximately scale invariant spectrum on cosmological scales.
The boundary for eternal inflation is roughly when the density fluctuations are
O(1), and this occurs when the fluctuations in the time delay are h(H δt)2 i = O(1).
To convert this into a lower bound on m0 , let us imagine that m0 is even smaller than
H. In this regime, the growth rate λ can be estimated using eq. (3.57) as λ ∼ m20 /H 2 .
Using eq. (3.33) this gives h(H δt)2 i ∼ H 4 /(N m40 ). This implies that eternal inflation
occurs when the waterfall mass is below a critical value mc0 , which is roughly
mc0 ∼
H
.
N 1/4
(3.60)
So when N ∼ 1 we cannot have m0 near H, because we then enter eternal inflation.
On the other hand, for large N we are allowed to have m0 near H and avoid this
problem. This makes sense intuitively, because for many fields it is statistically favorable for at least one of the fields to fall off the hill-top, causing inflation to end.
Hence large N is more easily compatible with the above set of constraints than low
N.
111
3.6
Discussion and Conclusions
In this work we studied density perturbations in hybrid inflation caused by N waterfall
fields, which contains a spike in the spectrum. We derived expressions for correlation
functions of the time-delay and constrained parameters with observations.
Density Perturbations: We derived a convergent series expansion in powers of
1/N and ∆(x), the dimensionless correlation function for the field, for the two-point
function of the time-delay for any N , and the leading order behavior of the threepoint function of the time-delay for large N . These correlation functions are well
approximated by the first term in the series for large N (even for N = 2 the leading
term is moderately accurate to ∼ 30%, and much more accurate for higher N ). In
this regime, the fluctuations are suppressed, with two-point and three-point functions
given by
∆2 (x)
,
2λ2 N
∆(|x − y|)∆(x)∆(y)
hδt(x) δt(y) δt(0)i ≈ −
.
λ3 N 2
hδt(x) δt(0)i ≈
(3.61)
Although this reduces the spike in the spectrum, for any moderate value of N , such
as N = 3, 4, 5, the amplitude of the spike is still quite large (orders of magnitude
larger than the ∼ 10−5 level fluctuations on larger scales relevant to the CMB), and
may have significant astrophysical consequences. Also, the relative size of the three√
point function to the 3/2 power of the two-point function scales as ∼ 1/ N . In
accordance with the central limit theorem, the fluctuations become more Gaussian at
large N . This will make the analysis of the subsequent cosmological evolution more
manageable, as this provides a simple spectrum for initial conditions. We note that
since we are considering small scales compared to the CMB, then this non-Gaussianity
evades Planck bounds [6].
Constraints: We mentioned that hybrid models avoid topological defects for large
N and satisfy constraints on tensor modes for any N . A very serious constraint on
hybrid models comes from the observed spectral index ns ≈ 0.96, which requires the
112
potential to flatten at large field values. One consequence of this is that the timer field
mass mψ needs to be only a little smaller than the Hubble parameter during inflation
H, or else the model is significantly fine tuned. For a large value of the waterfall field
mass m0 , this would imply a large growth rate of fluctuations, a rapid termination
of inflation, and in turn a density spike on very small scales. Otherwise, we need to
make the waterfall field mass m0 somewhat close to H, but this faces problems with
eternal inflation. However, by using a large number of waterfall fields N , it is safer
to make the waterfall field mass m0 somewhat close to H. This reduces the growth
rate of fluctuations, prolonging the waterfall phase for many e-foldings.
Thus large N presents a plausible setup to establish a spike in the density
perturbations on astrophysically large length scales that is consistent with other
constraints.
Outlook: It may be possible that these perturbations seed primordial black holes,
which may be relevant to supermassive black holes, or an intriguing form of dark
matter. An investigation into these topics is underway. It would be important to
fully explore the eternal inflation bound and the effects on the spectrum for relatively
light waterfall field masses. Finally, it would be of interest to try to embed these large
N models into fundamental physics, such as string theory, and to explore reheating
[36] and baryogenesis [37, 38] in this framework.
3.7
Acknowledgements
We would like to thank Victor Buza and Alexis Giguere for very helpful discussions.
This work is supported by the U.S. Department of Energy under cooperative research
agreement Contract Number DE-FG02-05ER41360 and in part by MIT’s Undergraduate Research Opportunities Program.
113
3.8
Appendix
3.8.1
Series Expansion to Higher Orders
Earlier we computed the two-point correlation function for the time-delay at quadratic
order, and then stating the results at all orders. Here we mention the results order
by order.
a. Cubic Order
At cubic order ∼ Φ3 we find
(2λ)2 hδt(x)δt(0)i3 = −
8∆2
N2
(3.62)
b. Quartic Order
At quartic order ∼ Φ4 we find
(2λ)2 hδt(x)δt(0)i4 =
8∆2 + 2∆4 40∆2 + 4∆4
+
N2
N3
(3.63)
c. Quintic Order
At quintic order ∼ Φ5 we find
96∆2 + 32∆4
N3
256∆2 + 64∆4
−
N4
(2λ)2 hδt(x)δt(0)i5 = −
d. Sextic Order
114
(3.64)
At sextic order ∼ Φ6 we find
(2λ)2 hδt(x)δτ (0)i6 =
168∆2 + 72∆4 + 16∆6
3N 3
2
1056∆ + 464∆4 + 32∆6
+
N4
2
6144∆ + 2496∆4 + 128∆6
+
3N 5
(3.65)
e. Septic Order
At septic order ∼ Φ7 we find
32(43∆2 + 22∆4 + 6∆6 )
4
N
1
+O
N5
(2λ)2 hδt(x)δt(0)i7 = −
(3.66)
f. Octic Order
At octic order ∼ Φ8 we find
(2λ)2 hδt(x)δt(0)i8 =
3.8.2
8(72∆2 + 39∆4 + 16∆6 + 3∆8 )
4
N
1
+O
N5
(3.67)
Two-Point Function for Even Number of Fields
When N is even the expression always involves the polylog function that we found
for N = 2, plus corrections that depend on N . We find that the form of the answer
is
PN (∆2 )
∆N −4
2
P̄N (∆ ) ln(1 − ∆2 )
+
,
∆N −2
(2λ)2 hδt(x)δt(0)i = Li2 (∆2 ) +
115
(3.68)
where
PN (∆2 ) is a polynomial of degree (N − 4)/2,
(3.69)
P̄N (∆2 ) is a polynomial of degree (N − 2)/2.
(3.70)
For N = 4, we find
P4 (∆2 ) = −1,
(3.71)
P̄4 (∆2 ) = −1 + ∆2 .
(3.72)
For N = 6, we find
1 7 2
− ∆,
2 4
3
1
− 2∆2 + ∆4 .
P̄6 (∆2 ) =
2
2
P6 (∆2 ) =
(3.73)
(3.74)
For N = 8, we find
1
P8 (∆2 ) = − +
3
1
P̄8 (∆2 ) = − +
3
3.8.3
4 2 85 4
∆ − ∆,
3
36
3 2
11
∆ − 3∆4 + ∆6 .
2
6
(3.75)
(3.76)
Alternative Derivation of Time-Delay Spectra
Here we write the two-point function as a multidimensional integral. It is convenient
now to switch to a vector notation thus making the components of φ explicit.
~ x ≡ (X1 , X2 , X3 , . . . , XN ),
φ̃(0, t) ≡ φ
(3.77)
~ 0 ≡ (XN +1 , XN +2 , XN +3 , . . . , X2N ),
φ̃(x, t) ≡ φ
(3.78)
~ ≡ (X1 , X2 , X3 , . . . , X2N ).
X
(3.79)
116
~ with probability distribution
The average value of a function F of a random variable X
function p(X) is given by
Z
hF [X]i =
dXp(X)F [X].
(3.80)
~ follows a joint Gaussian distriSince we are using a free field approximation, X
bution
1
1 T −1
p
p(X) =
exp − X Σ X ,
2
(2π)N det(Σ)
(3.81)
where
Σij = hXi Xj i,
(3.82)
is the correlation matrix. The components of Σ can be easily calculated using the
commutation relations for the creation and annihilation operators in φ(x, t), from
Eq. (3.11). Due to the high degree of symmetry the matrix itself has a very simple
structure:
Σij =
1
(δi,j + δi,(j±N ) ∆).
N
(3.83)
Finally, we can write the two-point function as:
(2λ)2 hδt(x)δt(0)i
Z
~
N 2 dX
~ 2
~ 2
=
N log(|φ0 | ) log(|φx | ) ×
N
2
(2π) (1 − ∆ ) 2
N
~ 0 |2 + |φ
~ x |2 − 2N ∆φ
~0 · φ
~ x ]}.
exp{−
[|φ
2
2(1 − ∆ )
(3.84)
Then we can Taylor expand the integrand in powers of ∆. We see that all odd
powers of ∆ vanish, leaving only even powers in the expansion. Performing these
integrals term by term in the Taylor expansion, leads to the results reported earlier
in eqs. (3.35, 3.36, 3.37).
117
118
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[35] A.H. Guth, “Quantum fluctuations in cosmology and how they lead to a multiverse,” to be published in the Proceedings of the 25th Solvay Conference in
Physics, The Theory of the Quantum World, Brussels, October 2011.
[36] L. Kofman, A. D. Linde and A. A. Starobinsky, “Towards the theory of reheating
after inflation,” Phys. Rev. D 56, 3258 (1997) [hep-ph/9704452].
[37] M. P. Hertzberg and J. Karouby, “Baryogenesis from the Inflaton Field,”
arXiv:1309.0007 [hep-ph].
[38] M. P. Hertzberg and J. Karouby, “Generating the Observed Baryon Asymmetry
from the Inflaton Field,” arXiv:1309.0010 [hep-ph].
122
Chapter 4
Primordial Bispectrum from
Multifield Inflation with
Nonminimal Couplings
Realistic models of high-energy physics include multiple scalar fields. Renormalization
requires that the fields have nonminimal couplings to the spacetime Ricci curvature
scalar, and the couplings can be large at the energy scales of early-universe inflation.
The nonminimal couplings induce a nontrivial field-space manifold in the Einstein
frame, and they also yield an effective potential in the Einstein frame with nontrivial
curvature. The ridges or bumps in the Einstein-frame potential can lead to primordial non-Gaussianities of observable magnitude. We develop a covariant formalism to
study perturbations in such models and calculate the primordial bispectrum. As in
previous studies of non-Gaussianities in multifield models, our results for the bispectrum depend sensitively on the fields’ initial conditions.
4.1
Introduction
Inflationary cosmology remains the leading account of the very early universe, consistent with high-precision measurements of the cosmic microwave background radiation
(CMB) [1, 2, 3]. A longstanding challenge, however, has been to realize success123
ful early-universe inflation within a well-motivated model from high-energy particle
physics.
Realistic models of high-energy physics routinely include multiple scalar fields
[4, 5]. Unlike single-field models, multifield models generically produce entropy (or
isocurvature) perturbations. The entropy perturbations, in turn, can cause the gaugeinvariant curvature perturbation, ζ, to evolve even on the longest length-scales, after
modes have been stretched beyond the Hubble radius during inflation [6, 7, 8, 9, 10,
11, 12, 13]. Understanding the coupling and evolution of entropy perturbations in
multifield models is therefore critical for studying features in the predicted power
spectrum, such as non-Gaussianities, that are absent in simple single-field models.
(For reviews see [14, 12, 13, 15, 16, 17].)
Recent reviews of primordial non-Gaussianities have emphasized four criteria, at
least one of which must be satisfied as a necessary (but not sufficient) condition for
observable power spectra to deviate from predictions of single-field models. These
criteria include [17, 15]: (1) multiple fields; (2) noncanonical kinetic terms; (3) violation of slow-roll; or (4) an initial quantum state for fluctuations different than the
usual Bunch-Davies vacuum. As we demonstrate here, the first three of these criteria
are generically satisfied by models that include multiple scalar fields with nonminimal
couplings to the spacetime Ricci curvature scalar.
Nonminimal couplings arise in the action as necessary renormalization counterterms for scalar fields in curved spacetime [18, 19, 22, 23, 20, 21]. In many models the
nonminimal coupling strength, ξ, grows without bound under renormalization-group
flow [21]. In such models, if the nonminimal couplings are ξ ∼ O(1) at low energies,
they will rise to ξ 1 at the energy scales of early-universe inflation. We therefore
expect realistic models of inflation to incorporate multiple scalar fields, each with a
large nonminimal coupling. (Non-Gaussianities in single-field models with nonminimal couplings have been studied in [24].)
Upon performing a conformal transformation to the Einstein frame — in which
the gravitational portion of the action assumes canonical Einstein-Hilbert form —
the nonminimal couplings induce a field-space manifold that is not conformal to
124
flat [25]. The curvature of the field-space manifold, in turn, can induce additional
interactions among the matter fields, beyond those included in the Jordan-frame
potential. Moreover, the scalar fields necessarily acquire noncanonical kinetic terms
in the Einstein frame. These new features can have a dramatic impact on the behavior
of the fields during inflation, and hence on the primordial power spectrum.
Chief among the multifield effects for producing new features in the primordial
power spectrum is the ability of fields’ trajectories to turn in field-space as the system
evolves. Such turns are not possible in single-field models, which include only a single
direction of field-space. In the case of multiple fields, special features in the effective potential, such as ridges or bumps, can focus the background fields’ trajectories
through field space or make them diverge. When neighboring trajectories diverge,
primordial bispectra can be amplified to sufficient magnitude that they should be
detectable in the CMB [26, 14, 30, 16, 15, 13, 32, 31, 27, 28, 29].
To date, features like ridges in the effective potential have been studied for the
most part phenomenologically rather than being strongly motivated by fundamental
physics. Here we demonstrate that ridges arise naturally in the Einstein-frame effective potential for models that incorporate multiple fields with nonminimal couplings.
Likewise, as noted above, models with multiple nonminimally coupled scalar fields
necessarily include noncanonical kinetic terms in the Einstein frame, stemming from
the curvature of the field-space manifold. Both the bumpy features in the potential
and the nonzero curvature of the field-space manifold routinely cause the fields’ evolution to depart from slow-roll for some duration of their evolution during inflation.
Recent analyses of primordial non-Gaussianities have emphasized two distinct
types of fine-tuning needed to produce observable bispectra: fine-tuning the shape
of the effective potential to include features like ridges; and separately fine-tuning
the fields’ initial conditions so that the fields begin at or near the top of these ridges
[30, 29, 32, 31]. Here we show that the first of these types of fine-tuning is obviated
for multifield models with nonminimal couplings; such features of the potential are
generic. The second type of fine-tuning, however, is still required: even in the presence
of ridges and bumps, the fields’ initial conditions must be fine-tuned in order to
125
produce measurable non-Gaussianities.
In Section II we examine the evolution of the fields in the Einstein frame and emphasize the ubiquity of features such as ridges that could make the fields’ trajectories
diverge in field space. Section III introduces our covariant, multifield formalism for
studying the evolution of background fields and linearized perturbations on the curved
field-space manifold. In Section IV we analyze adiabiatic and entropy perturbations
and quantify their coupling using a covariant version of the familiar transfer-function
formalism [11, 13, 35]. In Section V we build on recent work [36, 37, 38] to calculate the primordial bispectrum for multifield models, applying it here to models with
nonminimal couplings. We find that although the nonminimal couplings induce new
interactions among the entropy perturbations compared to models in which all fields
have minimal coupling, the dominant contribution to the bispectrum remains the familiar local form of fN L , made suitably covariant to apply to the curved field-space
manifold. Concluding remarks follow in Section VI. We collect quantities relating to
the curvature of the field-space manifold in the Appendix.
4.2
Evolution in the Einstein Frame
We consider N scalar fields in (3 + 1) spacetime dimensions, with spacetime metric signature (−, +, +, +). We work in terms of the reduced Planck mass, Mpl ≡
(8πG)−1/2 = 2.43 × 1018 GeV. Greek letters label spacetime indices, µ, ν = 0, 1, 2, 3;
lower-case Latin letters label spatial indices, i, j = 1, 2, 3; and upper-case Latin letters
label field-space indices, I, J = 1, 2, ..., N .
In the Jordan frame, the scalar fields’ nonminimal couplings to the spacetime Ricci
curvature scalar remain explicit in the action. We denote quantities in the Jordan
frame with a tilde, such as the spacetime metric, g̃µν (x). The action for N scalar
fields in the Jordan frame may be written
Z
SJordan =
p
1
I
µν
I
J
I
d x −g̃ f (φ )R̃ − G̃IJ g̃ ∂µ φ ∂ν φ − Ṽ (φ ) ,
2
4
126
(4.1)
where f (φI ) is the nonminimal coupling function and Ṽ (φI ) is the potential for the
scalar fields in the Jordan frame. We have included the possibility that the scalar fields
in the Jordan frame have noncanonical kinetic terms, parameterized by coefficients
G̃IJ (φK ). Canonical kinetic terms correspond to G̃IJ = δIJ .
We next perform a conformal transformation to work in the Einstein frame, in
which the gravitational portion of the action assumes Einstein-Hilbert form. We
define a rescaled spacetime metric tensor, gµν (x), via the relation,
gµν (x) = Ω2 (x) g̃µν (x),
(4.2)
where the conformal factor is related to the nonminimal coupling function as
Ω2 (x) =
2
f (φI (x)).
Mpl2
(4.3)
Eq. (4.1) then takes the form [25]
Z
SEinstein =
2
Mpl
√
1
µν
I
J
I
d x −g
R − GIJ g ∂µ φ ∂ν φ − V (φ ) .
2
2
4
(4.4)
The potential in the Einstein frame is scaled by the conformal factor,
V (φI ) =
Mpl4
1
I
Ṽ
(φ
)
=
Ṽ (φI ).
Ω4 (x)
4f 2 (φI )
(4.5)
The coefficients of the noncanonical kinetic terms in the Einstein frame depend on
the nonminimal coupling function, f (φI ), and its derivatives, and are given by [39, 25]
Mpl2
3
K
GIJ (φ ) =
G̃IJ (φ ) +
f,I f,J ,
2f (φI )
f (φI )
K
(4.6)
where f,I = ∂f /∂φI .
As demonstrated in [21], the nonminimal couplings induce a field-space manifold
in the Einstein frame, associated with the metric GIJ (φK ) in Eq. (4.6), which is not
conformal to flat for models in which multiple scalar fields have nonminimal cou127
plings in the Jordan frame. Thus there does not exist any combination of conformal
transformation plus field-rescalings that can bring the induced metric into the form
GIJ = δIJ . In other words, multifield models with nonminimal couplings necessarily include noncanonical kinetic terms in the Einstein frame, even if the fields have
canonical kinetic terms in the Jordan frame, G̃IJ = δIJ . When analyzing multifield
inflation with nonminimal couplings, we therefore must work either with a noncanonical gravitational sector or with noncanonical kinetic terms. Here we adopt the latter.
Because there is no way to avoid noncanonical kinetic terms in the Einstein frame in
such models, we do not rescale the fields. For the remainder of the chapter, we restrict
attention to models with canonical kinetic terms in the Jordan frame, G̃IJ = δIJ , in
which the curvature of the field-space manifold in the Einstein frame depends solely
upon f (φI ) and its derivatives.
Varying the action of Eq. (4.4) with respect to gµν (x) yields the Einstein field
equations,
1
1
Rµν − gµν R = 2 Tµν ,
2
Mpl
(4.7)
where
I
J
Tµν = GIJ ∂µ φ ∂ν φ − gµν
1
αβ
I
J
I
GIJ g ∂α φ ∂β φ + V (φ ) .
2
(4.8)
Varying Eq. (4.4) with respect to φI yields the equation of motion,
φI + g µν ΓIJK ∂µ φJ ∂ν φK − G IK V,K = 0,
(4.9)
where φI ≡ g µν φI;µ;ν and ΓIJK (φL ) is the Christoffel symbol for the field-space
manifold, calculated in terms of GIJ .
We expand each scalar field to first order around its classical background value,
φI (xµ ) = ϕI (t) + δφI (xµ ),
(4.10)
and also expand the scalar degrees of freedom of the spacetime metric to first order,
perturbing around a spatially flat Friedmann-Robertson-Walker (FRW) metric [40,
128
11, 12],
ds2 = gµν (x) dxµ dxν
(4.11)
= − (1 + 2A) dt2 + 2a (∂i B) dxi dt + a2 [(1 − 2ψ)δij + 2∂i ∂j E] dxi dxj ,
where a(t) is the scale factor. To background order, the 00 and ij components of Eq.
(4.7) may be combined to yield the usual dynamical equations,
1
1
I J
I
GIJ ϕ̇ ϕ̇ + V (ϕ ) ,
H =
3Mpl2 2
1
Ḣ = −
GIJ ϕ̇I ϕ̇J ,
2Mpl2
2
(4.12)
where H ≡ ȧ/a is the Hubble parameter, and the field-space metric is evaluated at
background order, GIJ = GIJ (ϕK ).
Both the curvature of the field-space manifold and the form of the effective potential in the Einstein frame depend upon the nonminimal coupling function, f (φI ).
The requirement of renormalizability for scalar matter fields in a (classical) curved
background spacetime dictates the form of f (φI ) [18, 19, 20, 21]:
"
#
X
1
2
f (φI ) =
M02 +
ξI φI
,
2
I
(4.13)
where M0 is some mass-scale that could be distinct from Mpl , and the nonminimal
couplings ξI are dimensionless constants that need not be equal to each other. If any
of the fields develop nonzero vacuum expectation values, hφI i = v I , then one may
P
expect Mpl2 = M02 + I ξI (v I )2 . Here we will assume either that v I = 0 for each field
√
or that ξI v I Mpl , so that M0 ' Mpl .
The nonminimal couplings ξI could in principle take any “bare” value. (Conformal
couplings correspond to ξI = −1/6; we only consider positive couplings here, ξI >
0.) Under renormalization-group flow the constants vary logarithmically with energy
scale. The exact form of the β functions depends upon details of the matter sector,
but for models whose content is akin to the Standard Model the β functions are
129
positive and the flow of ξI has no fixed point, rising with energy scale without bound
[21]. Studies of the flow of ξ in the case of Higgs inflation [41] indicate growth of ξ
by O(101 − 102 ) between the electroweak symmetry-break scale, Λ ∼ 102 GeV, and
typical inflationary scales, Λ ∼ 1016 GeV [42]. Hence we anticipate that realistic
models will include nonminimal couplings ξI 1 during inflation.
Renormalizable potentials in (3+1) spacetime dimensions can include terms up to
quartic powers of the fields. A potential in the Jordan frame that assumes a generic
renormalizable, polynomial form such as
Ṽ (φI ) =
2 J 2 1 X
4
1 X 2 I 2 1 X
mI φ +
gIJ φI
+
λI φI
φ
2 I
2 I<J
4 I
(4.14)
will yield an effective potential in the Einstein frame that is stretched by the conformal
factor in accord with Eq. (4.5). As the Jth component of φI becomes arbitrarily large
the potential in that direction will become asymptotically flat,
V (φI ) =
Mpl4 λJ
Mpl4 Ṽ (φI )
→
4 f 2 (φI )
4 ξJ2
(4.15)
(no sum on J), unlike the quartic behavior of the potential in the large-field limit in
the Jordan frame. (The flatness of the effective potential for large field values was
one inspiration for Higgs inflation [41].) Inflation in such models occurs in a regime of
field values such that ξJ (ϕJ )2 Mpl2 for at least one component, J. As emphasized in
[41], for large nonminimal couplings, ξJ 1, all of inflation therefore may occur for
field values that satisfy |ϕJ | < Mpl , unlike the situation for ordinary chaotic inflation
with polynomial potentials and minimal couplings.
Although the effective potential in the Einstein frame will asymptote to a constant
value in any given direction of field space, the constants will not, in general, be equal
to each other. Thus at finite values of the fields, the potential will generically develop
features, such as ridges or bumps, that are absent from the Jordan-frame potential.
Because the asymptotic values of V (φI ) in any particular direction are proportional
to 1/ξJ2 , the steepness of the ridges depends sharply on the ratios of the nonminimal
130
coupling constants. If some explicit symmetry, such as the SU (2) electroweak gauge
symmetry obeyed by the Higgs multiplet in Higgs inflation [41], forces all the couplings
to be equal — ξI = ξ, m2I = m2 , and λI = gIJ = λ for all I, J — then the ridges
in the Einstein-frame potential disappear and the potential asymptotes to the same
constant value in each direction of field space. We study the dynamics of such special
cases in [43]. For the remainder of this chapter, we consider models in which the
constants are of similar magnitude but not exactly equal to each other.
For definiteness, consider a two-field model with a potential in the Jordan frame
of the form
1
1
1
λφ
λχ
Ṽ (φ, χ) = m2φ φ2 + m2χ χ2 + gφ2 χ2 + φ4 + χ4
2
2
2
4
4
(4.16)
and nonminimal coupling function given by
f (φ, χ) =
1 2
Mpl + ξφ φ2 + ξχ χ2 .
2
(4.17)
In the Einstein frame the potential becomes
Mpl4 2m2φ φ2 + 2m2χ χ2 + 2gφ2 χ2 + λφ φ4 + λχ χ4
.
V (φ, χ) =
2
2
4
Mpl + ξφ φ2 + ξχ χ2
(4.18)
See Fig. 4-1.
In addition to the ridges shown in Fig. 4-1, other features of the Einstein-frame
potential can arise depending on the Jordan-frame couplings. For example, the tops
of the ridges can develop small indentations, such that the top of a ridge along χ ∼ 0
becomes a local minimum rather than a local maximum. In that case, field trajectories
that begin near the top of a ridge tend to focus rather than diverge, keeping the
amplitude of non-Gaussianities very small. For the two-field potential of Eq. (4.18),
we find [44]
∂χ2 V
|χ=0
=
1
Mpl2 + ξφ φ2
2
4
2
2
2
2
2
3 (gξφ − λφ ξχ ) φ + ξφ mχ − 2ξχ mφ + gMpl φ + mχ Mpl .
(4.19)
131
Figure 4-1: The Einstein-frame effective potential, Eq. (4.18), for a two-field model. The
potential shown here corresponds to the couplings ξχ /ξφ = 0.8, λχ /λφ = 0.3, g/λφ = 0.1,
2.
and m2φ = m2χ = 10−2 λφ Mpl
For realistic values of the masses that satisfy m2φ , m2χ Mpl2 , and at early times
when ξφ φ2 Mpl2 , the top of the ridge along the χ ∼ 0 direction will remain a local
maximum if
gξφ < λφ ξχ .
(4.20)
When the couplings satisfy Eq. (4.20), the shape of the potential in the vicinity of
2
its ridges is similar to that of the product potential, V = m2 e−λφ χ2 , which has been
studied in detail in [30, 32]. Trajectories of the fields that begin near each other close
to the top of a ridge will diverge as the system evolves; that divergence in trajectories
can produce a sizeable amplitude for the bispectrum, as we will see below.
Even potentials with modest ratios of the nonminimal couplings can produce trajectories that diverge sharply, as shown in Fig. 4-2. As we will see in Section V,
trajectory 2 of Fig. 4-2 (solid red line) yields a sizeable amplitude for the bispectrum
that is consistent with present bounds, whereas trajectories 1 and 3 produce negligible non-Gaussianities. We will return to the three trajectories of Fig. 2 throughout
the chapter, as illustrations of the types of field dynamics that yield interesting possibilities for the power spectrum.
Unlike the product potential studied in [30, 32], the potential of Eq. (4.18) con132
Figure 4-2: Parametric plot of the fields’ evolution superimposed on the Einstein-frame
potential. Trajectories for the fields φ and χ that begin near the top of a ridge will diverge.
In this case, the couplings of the potential are ξφ = 10, ξχ = 10.02, λχ /λ
pφ = 0.5, g/λφ = 1,
and mφ = mχ = 0. (We use a dimensionless time variable, τ ≡
λφ Mpl t, so that
the Jordan-frame couplings are measured in units of λφ .) The trajectories shown here
each have the initial condition φ(τ0 ) = 3.1 (in units of Mpl ) and different values of χ(τ0 ):
χ(τ0 ) = 1.1 × 10−2 (“trajectory 1,” yellow dotted line); χ(τ0 ) = 1.1 × 10−3 (“trajectory 2,”
red solid line); and χ(τ0 ) = 1.1 × 10−4 (“trajectory 3,” black dashed line).
133
Figure 4-3: The evolution of the Hubble parameter (black dashed line) and the background
fields, φ(τ ) (red solid line) and χ(τ ) (blue dotted line), for trajectory 2 of Fig. 2. (We use
the same units as in Fig. 2, and have plotted 100H so its scale is commensurate with the
magnitude of the fields.) For these couplings and initial conditions the fields fall off the
ridge in the potential at τ = 2373 or N = 66.6 efolds, after which the system inflates for
another 4.9 efolds until τend = 2676, yielding Ntotal = 71.5 efolds.
tains valleys in which the system will still inflate. For trajectories 1 (orange dotted
line) and 2 (red solid line) in Fig. 4-2, for example, the system begins near χ ∼ 0
and rolls off the ridge; because λχ /ξχ2 6= 0, the valleys in the χ direction are also false
vacua and hence the system continues to inflate as the fields relax toward the global
minimum at φ = χ = 0. Near the end of inflation, when ξφ φ2 + ξχ χ2 < Mpl2 , the
fields oscillate around the global minimum of the potential, which can drive a period
of preheating. See Fig. 4-3.
Evolution of the fields like that shown in Fig. 4-3 is generic for this class of
models when the fields begin near the top of a ridge, and can produce interesting
phenomenological features in addition to observable bispectra. For example, the
oscillations of φ around φ = 0 when the system first rolls off the ridge could produce
an observable time-dependence of the scale factor during inflation, as analyzed in
[45]. The added period of inflation from the false vacuum of the χ valley could lead
to scale-dependent features in the power spectrum associated with double-inflation
[46].
134
Figure 4-4: Models with nonzero masses include additional features in the Einstein-frame
potential which can also cause neighboring field trajectories to diverge. In this case, we
superimpose the evolution of the fields φ and χ on the Einstein-frame potential. The
2 and
parameters shown here are identical to those in Fig. 4-2 but with m2φ = 0.075 λφ Mpl
2 rather than 0. The initial conditions match those of trajectory 3 of
m2χ = 0.0025 λφ Mpl
Fig. 4-2: φ(τ0 ) = 3.1 and χ(τ0 ) = 1.1 × 10−4 in units of Mpl .
In the class of models we consider here, neighboring trajectories may also diverge
if we include small but nonzero bare masses for the fields. For example, in Fig. 4-4
we show the evolution of the fields for the same initial conditions as trajectory 3 of
Fig. 4-2 — the black, dashed curve that barely deviates from the middle of the ridge.
The evolution shown in Fig. 4-2 was for the case mφ = mχ = 0. If, instead, we
include nonzero masses, then the curvature of the effective potential at small field
values becomes different from the zero-mass case. In particular, for positive, real
values of the masses, the ridges develop features that push the fields off to one side,
recreating behavior akin to what we found in trajectories 1 and 2 of Fig. 4-2.
Because the field-space manifold is curved, the fields’ trajectories will turn even
in the absence of tree-level couplings from the Jordan-frame potential: the fields’
geodesic motion alone is nontrivial. The Ricci scalar for the field-space manifold in
the two-field case is given in Eq. (4.115). In Fig. 4-5 we plot the fields’ motion in
the curved manifold for the case when Ṽ (φ, χ) = V (φ, χ) = 0. The curvature of the
manifold is negligible at large field values but grows sharply near φ ∼ χ ∼ 0.
135
Figure 4-5: Parametric plot of the evolution of the fields φ and χ superimposed on the
Ricci curvature scalar for the field-space manifold, R, in the absence of a Jordan-frame
potential. The fields’ geodesic motion is nontrivial because of the nonvanishing curvature.
Shown here is the case ξφ = 10, ξχ = 10.02, φ(τ0 ) = 0.75, χ(τ0 ) = 0.01, φ0 (τ0 ) = −0.01, and
χ0 (τ0 ) = 0.005.
136
Given the nonvanishing curvature of the field-space manifold, we must study the
evolution of the fields and their perturbations with a covariant formalism, to which
we now turn.
4.3
Covariant Formalism
A gauge-invariant formalism for studying perturbations in multifield models in the
Jordan frame was developed in [47, 48]. In this chapter we work in the Einstein
frame, following the approach established in [7, 8, 9, 10, 27, 29, 30, 31, 28, 35, 33,
34, 32, 36, 37, 38]. Our approach is especially indebted to the geometric formulation
of [32]. In [32], the authors introduce a particular tetrad construction with which to
label the field-space manifold locally, which they dub the “kinematical basis.” The
adoption of the kinematical basis simplifies certain expressions and highlights features
of physical interest in the primordial power spectrum, but it does so at the expense of
obscuring the relationship between observable quantities and the fields that appear
in the original Lagrangian, in terms of which any given model is specified. Rather
than adopt the kinematical basis here, we develop a covariant approach in terms of
a single coordinate chart that covers the entire field manifold. This offers greater
insight into the global structure of the manifold, as illustrated in Fig. 4-5. We also
keep coordinate labels explicit, which facilitates application of our formalism to the
original basis of fields, φI , that appears in the governing Lagrangian. Also unlike [32],
we work in terms of cosmic time, t, rather than the number of efolds during inflation,
N , because we are interested in applying our formalism (in later work) to eras such
as preheating, for which N is a poor dynamical parameter. Because of these formal
distinctions from [32], we briefly introduce our general formalism in this section.
We expand each scalar field to first order around its classical background value,
as in Eq. (4.10). The background fields, ϕI (t), parameterize classical paths through
the curved field-space manifold, and hence can be used as coordinate descriptions of
the trajectories. Just like spacetime coordinates in general relativity, xµ , the array ϕI
is not a vector in the field-space manifold [49]. Infinitesimal displacements, dϕI , do
137
behave as proper vectors, and hence so do derivatives of ϕI with respect to an affine
parameter such as t.
For any vector in the field space, AI , we define a covariant derivative with respect
to the field-space metric as usual by
DJ AI = ∂J AI + ΓIJK AK .
(4.21)
Following [?, ?, 18], we also introduce a covariant derivative with respect to cosmic
time via the relation
Dt AI ≡ ϕ̇J DJ AI = ȦI + ΓIJK AJ ϕ̇K ,
(4.22)
where overdots denote derivatives with respect to t. The construction of Eq. (4.22)
is essentially a directional derivative along the trajectory.
For models with nontrivial field-space manifolds, the tangent space to the manifold
at one time will not coincide with the tangent space at some later time. Hence
the authors of [36, 37] introduce a covariant means of handling field fluctuations,
which we adopt here. As specified in Eq. (4.10), the value of the physical field at
a given location in spacetime, φI (xµ ), consists of the homogenous background value,
ϕI (t), and some gauge-dependent fluctuation, δφI (xµ ). The fluctuation δφI represents
a finite coordinate displacement from the classical trajectory, and hence does not
transform covariantly. This motivates a construction of a vector QI to represent the
field fluctuations in a covariant manner. The two field values, φI and ϕI , may be
connected by a geodesic in the field-space manifold parameterized by some parameter
λ, such that φI (λ = 0) = ϕI and φI (λ = 1) = ϕI + δφI . These boundary conditions
allow us to identify a unique vector, QI , that connects the two field values, such that
Dλ φI |λ=0 = QI . One may then expand δφI in a power series in QI [36, 37],
δφI = QI −
1 I
1 I
Γ JK QJ QK +
Γ LM ΓMJK − ΓIJK,L QJ QK QL + ...
2!
3!
(4.23)
where the Christoffel symbols are evaluated at background order in the fields, ΓIJK =
138
ΓIJK (ϕL ). To first order in fluctuations δφI → QI , and hence at linear order we
may treat the two quantities interchangeably. When we consider higher-order combinations of the field fluctuations below, however, such as the contribution of the
three-point function of field fluctuations to the bispectrum, we must work in terms
of the vector QI rather than δφI .
We introduce the gauge-invariant Mukhanov-Sasaki variables for the perturbations
[40, 11, 12],
ϕ̇I
Q ≡ Q + ψ.
H
I
I
(4.24)
Because both QI and ϕ̇I are vectors in the field-space manifold, QI is also a vector.
The Mukhanov-Sasaki variables, QI , should not be confused with the vector of field
fluctuations, QI . The QI are gauge-invariant with respect to spacetime gauge transformations up to first order in the perturbations, and are constructed from a linear
combination of field fluctuations and metric perturbations. The quantity QI does
not incorporate metric perturbations; it is constructed from the (gauge-dependent)
field fluctuations and background-order quantities such as the field-space Christoffel
symbols. At lowest order in perturbations, QI → QI in the spatially flat gauge.
Using Eq. (4.24), Eq. (4.9) separates into background and first-order expressions,
Dt ϕ̇I + 3H ϕ̇I + G IK V,K = 0,
(4.25)
and
"
Dt2 QI + 3HDt QI +
2
k I
1
δ J + MIJ − 2 3 Dt
2
a
Mpl a
3
a I
ϕ̇ ϕ̇J
H
#
QJ = 0.
(4.26)
The mass-squared matrix appearing in Eq. (4.26) is given by
MIJ ≡ G IK (DJ DK V ) − RILM J ϕ̇L ϕ̇M ,
(4.27)
where RILM J is the Riemann tensor for the field-space manifold. All expressions in
Eqs. (4.25), (4.26), and (4.27) involving GIJ , ΓIJK , RILM J , and V are evaluated at
139
background order in the fields, ϕI .
The system simplifies further if we distinguish between the adiabatic and entropic
directions in field space [7]. The length of the velocity vector for the background
fields is given by
|ϕ̇I | ≡ σ̇ =
p
GIJ ϕ̇I ϕ̇J .
(4.28)
Introducing the unit vector,
σ̂ I ≡
ϕ̇I
,
σ̇
(4.29)
the background equations, Eqs. (4.12) and (4.25), simplify to
1
1 2
σ̇ + V ,
H =
3Mpl2 2
1 2
Ḣ = −
σ̇
2Mpl2
(4.30)
σ̈ + 3H σ̇ + V,σ = 0,
(4.31)
V,σ ≡ σ̂ I V,I .
(4.32)
2
and
where we have defined
The background dynamics of Eqs. (4.30) and (4.31) take the form of a single-field
model with canonical kinetic term, with the exception that V (ϕI ) in Eqs. (4.30) and
(4.31) depends on all N independent fields, ϕI .
The directions in field space orthogonal to σ̂ I are spanned by
ŝIJ ≡ G IJ − σ̂ I σ̂ J .
140
(4.33)
The quantities σ̂ I and ŝIJ obey the useful relations
σ̂I σ̂ I = 1,
ŝIJ ŝIJ = N − 1,
(4.34)
ŝIA ŝAJ = ŝIJ ,
σ̂I ŝIJ = 0 for all J.
Therefore we may use σ̂ I and ŝIJ as projection operators to decompose any vector in
field space into components along the direction σ̂ I and perpendicular to σ̂ I as
AI = σ̂ I σ̂J AJ + ŝIJ AJ .
(4.35)
In particular, Ṡ I ≡ ŝIJ ϕ̇J vanishes identically, Ṡ I = 0. Thus all of the dynamics of
the background fields are captured by the behavior of σ̇ and σ̂ I .
Given the simple structure of the background evolution, Eqs. (4.30) and (4.31),
we introduce slow-roll parameters akin to the single-field case. We define
≡−
Ḣ
3σ̇ 2
=
H2
(σ̇ 2 + 2V )
(4.36)
and
ησσ ≡ Mpl2
Mσσ
,
V
(4.37)
where we have defined
MσJ ≡ σ̂I MIJ = σ̂ K (DK DJ V ) ,
(4.38)
Mσσ ≡ σ̂I σ̂ J MIJ = σ̂ K σ̂ J (DK DJ V ) .
The term in MIJ involving RILM J vanishes when contracted with σ̂I or σ̂ J due to
the first Bianchi identity (since the relevant term is already contracted with σ̂ L σ̂ M ),
and hence Mσσ is independent of RILM J . For trajectory 2 of Fig. 4-2 (solid red line),
we see that slow-roll is temporarily violated when the fields roll off the ridge of the
potential. See Fig. 4-6.
141
Figure 4-6: The slow-roll parameters (blue dashed line) and |ησσ | (solid red line) versus
N∗ for trajectory 2 of Fig. 2, where N∗ is the number of efolds before the end of inflation.
Note that |ησσ | temporarily grows significantly larger than 1 after the fields fall off the ridge
in the potential at around N∗ ∼ 5.
A central quantity of interest is the turn-rate [32], which we denote ω I . The
turn-rate is given by the (covariant) rate of change of the unit vector, σ̂ I ,
1
ω I ≡ Dt σ̂ I = − V,K ŝIK ,
σ̇
(4.39)
where the last expression follows upon using the equations of motion, Eqs. (4.25) and
(4.31). Because ω I ∝ ŝIK , we have
ω I σ̂I = 0.
(4.40)
Using Eqs. (4.33) and (4.39), we also find
Dt ŝIJ = −σ̂ I ω J − ω I σ̂ J .
(4.41)
For evolution of the fields like that shown in Fig. 4-2, the turn-rate peaks when the
fields roll off the ridge; see Fig. 4-7.
We may decompose the perturbations along directions parallel to and perpendic142
Figure 4-7: The turn-rate, ω = |ω I |, for the three trajectories of Fig. 4-2: trajectory
1 (orange dotted line); trajectory 2 (red solid line); and trajectory 3 (black dashed line).
The rapid oscillations in ω correspond to oscillations of φ in the lower false vacuum of the
χ valley. For trajectory 1, ω peaks at N∗ = 34.5 efolds before the end of inflation; for
trajectory 2, ω peaks at N∗ = 4.9 efolds before the end of inflation; and for trajectory 3, ω
remains much smaller than 1 for the duration of inflation.
ular to σ̂ I :
Qσ ≡ σ̂I QI ,
(4.42)
δsI ≡ ŝIJ QJ .
Note that δsI may be defined either in terms of the field fluctuations or the MukhanovSasaki variables, since ŝIJ δφJ = ŝIJ QJ . Though δsI is a vector in field-space with
N components, only N − 1 of these components are linearly independent. We will
isolate particular components of interest in Section IV.
Taking a Fourier transform, such that for any function F (t, xi ) we have a2 (t)∂i ∂ i F (t, xi ) =
−k 2 Fk (t) where k is the comoving wavenumber, Eq. (4.26) separates into two equations of motion (we suppress the label k on Fourier modes),
"
#
a3 σ̇ 2
Qσ
H
!
V,σ Ḣ
+
ωJ δsJ
σ̇
H
k2
1 d
Q̈σ + 3H Q̇σ + 2 + Mσσ − ω 2 − 2 3
a
Mpl a dt
d
=2
ωJ δsJ − 2
dt
143
(4.43)
and
Dt2 δsI
2
k I
σ̈
I
I
I
I
I
+ 3Hδ J + 2σ̂ ωJ Dt δs + 2 δ J + M J − 2σ̂ MσJ + ωJ
δsJ
a
σ̇
"
#
(4.44)
σ̈
Ḣ
I
= −2ω Q̇σ + Qσ − Qσ .
H
σ̇
Although the effective mass of the adiabatic perturbations, m2eff = Mσσ − ω 2 , is
independent of RILM J , the curvature of the field-space manifold introduces couplings
among components of the entropy perturbations, δsI , by means of the MIJ term in
Eq. (4.44). The quantities Qσ and (ωJ δsJ ) are scalars in field space, so the covariant
time derivatives in Eq. (4.43) reduce to ordinary time derivatives.
From Eqs. (4.43) and (4.44), it is clear that the adiabatic and entropy perturbations decouple if the turn-rate vanishes, ω I = 0. Moreover, Eq. (4.43) for Qσ is
identical in form to that of a single-field model (with m2eff = Mσσ − ω 2 ), but with a
nonzero source term that depends on the combination ωJ δsJ . Even in the presence
of significant entropy perturbations, δsI , the power spectrum for adiabatic perturbations will be devoid of features such as non-Gaussianities unless the turn-rate is
nonzero, ω I 6= 0.
4.4
Adiabatic and Entropy Perturbations
In Section III we identified the vector of entropy perturbations, δsI , which includes
N − 1 physically independent degrees of freedom. As we will see in this section, these
N − 1 physical components may be further clarified by introducing a particular set of
unit vectors and projection operators in addition to σ̂ I and ŝIJ . With them we may
identify components of δsI of particular physical interest.
We denote the gauge-invariant curvature perturbation as Rc , not to be confused
with the Ricci scalar for the field-space manifold, R. The perturbation Rc is defined
as [40, 11, 12]
Rc ≡ ψ −
H
δq,
(ρ + p)
144
(4.45)
where ρ and p are the background-order energy density and pressure for the fluid
filling the FRW spacetime, and δq is the energy-density flux of the perturbed fluid,
T 0i ≡ ∂i δq. Given Eq. (4.8), we find
1
ρ = σ̇ 2 + V,
2
1
p = σ̇ 2 − V,
2
(4.46)
δq = −GIJ ϕ̇I δφJ = −σ̇σ̂J δφJ ,
and hence, upon using Eqs. (4.24) and (4.42),
Rc = ψ +
H
H
σ̂J δφJ = Qσ .
σ̇
σ̇
(4.47)
We thus find that Rc ∝ Qσ , and that the righthand side of Eq. (4.44) is proportional
to Ṙc . Recall that these expressions hold to first order in fluctuations, for which
δφI → QI .
In the presence of entropy perturbations, the gauge-invariant curvature perturbation need not remain conserved, Ṙc 6= 0. In particular, the nonadiabatic pressure
perturbation is given by [11, 12]
ṗ
2V,σ
δpnad ≡ δp − δρ = −
δρm + 2σ̇ ωJ δsJ ,
ρ̇
3H σ̇
(4.48)
where δρm ≡ δρ − 3Hδq is the gauge-invariant comoving density perturbation. The
perturbed Einstein field equations (to linear order) require [11, 12]
δρm = −2Mpl2
k2
Ψ,
a2
(4.49)
where Ψ is the gauge-invariant Bardeen potential [40, 11, 12]
B
.
Ψ ≡ ψ + a H Ė −
a
2
(4.50)
Therefore in the long-wavelength limit, for k aH, the only source of nonadiabatic
145
pressure comes from the entropy perturbations, δsI . Using the usual relations [11, 12]
among the gauge-invariant quantities Rc and ζ ≡ −ψ + (H/ρ̇)δρ, we find
Ṙc =
H k2
2H
J
ω
δs
.
Ψ
+
J
σ̇
Ḣ a2
(4.51)
Thus even for modes with k aH, Rc will not be conserved in the presence of
entropy perturbations if the turn-rate is nonzero, ω I 6= 0.
Eqs. (4.43) and (4.51) indicate that a particular component of the vector δsI is of
special physical relevance: the combination (ωJ δsJ ) serves as the source for Qσ and
hence for Ṙc . Akin to the “kinematical basis” of [32], we may therefore introduce a
new unit vector that points in the direction of the turn-rate, ω I , together with a new
projection operator that picks out the subspace perpendicular to both σ̂ I and ω I :
ŝI ≡
γ
IJ
ωI
,
ω
≡G
IJ
(4.52)
I
J
I J
− σ̂ σ̂ − ŝ ŝ ,
where ω = |ω I | is the magnitude of the turn-rate vector. Using the relations in Eq.
(4.34), the definitions in Eq. (4.52) imply
ŝIJ = ŝI ŝJ + γ IJ ,
γ IJ γIJ = N − 2,
IJ
(4.53)
I
ŝ ŝJ = ŝ ,
σ̂I ŝI = σ̂I γ IJ = ŝI γ IJ = 0 for all J.
We then find
Dt ŝI = −ωσ̂ I − ΠI ,
(4.54)
Dt γ IJ = ŝI ΠJ + ΠI ŝJ
where
ΠI ≡
1
MσK γ IK ,
ω
146
(4.55)
and hence, from Eq. (4.53),
σ̂I ΠI = ŝI ΠI = 0.
(4.56)
The vector of entropy perturbations, δsI , may then be written as
δsI = ŝI Qs + B I ,
(4.57)
where
Qs ≡ ŝJ QJ ,
(4.58)
B I ≡ γ IJ QJ .
The quantity that sources Qσ and Rc is now easily identified as the scalar, ωJ δsJ =
ωQs , which corresponds to just one component of the vector δsI .
Making use of Eqs. (4.30), (4.47), and (4.51), the equation of motion for δsI in
Eq. (4.44) separates into
k2
2
2
Q̈s + 3H Q̇s + 2 + Mss + 3ω − Π Qs
a
ω k2
J
J
J
J
= 4Mpl2
Ψ
−
D
Π
B
−
Π
D
B
−
M
B
−
3H
Π
B
t
J
J
t
sJ
J
σ̇ a2
(4.59)
and
Dt2 B I + 3Hδ IJ + 2 σ̂ I ωJ − ŝI ΠJ Dt B J
2
k I
IA
I
I
+ 2 δ J + γ MAJ − σ̂ MσJ − ŝ (3HΠJ + Dt ΠJ ) B J
a
= 2ΠI Q̇s − γ IA MAs Qs + 3HΠI + Dt ΠI Qs .
(4.60)
In analogy to (4.38), we have introduced the projections
MsJ ≡ ŝI MIJ ,
Mss ≡ ŝI ŝJ MIJ .
147
(4.61)
Note, however, that unlike MσJ , the term in MIJ proportional to RILM J does not
vanish upon contracting with ŝI or ŝJ . Hence the Riemann-tensor term in MIJ
induces interactions among the components of δsI .
For models with N ≥ 3 scalar fields, we may introduce additional unit vectors
and projection operators with which to characterize components of B I . The next in
the series are given by
ûI ≡
q
IJ
ΠI
,
Π
≡γ
IJ
(4.62)
I J
− û û .
Repeating steps as before, we find
Dt ûI = ΠŝI + τ I ,
(4.63)
Dt q IJ = −ûI τ J − τ I ûJ ,
where
IK
1
σ̇ A
L
τ ≡
MsK + σ̂ σ̂L DA M K q .
Π
ω
(4.64)
B I = ûI Qu + C I
(4.65)
I
We then have
in terms of
Qu ≡ ûJ QJ ,
(4.66)
C I ≡ q IJ QJ .
This decomposition reproduces the structure in the “kinematical basis” [32] but can
be applied in any coordinate basis for the field-space manifold: Qσ is sourced by
Qs ; Qs is sourced by Qσ and Qu (though we have used Eq. (4.51) to substitute
the dependence on Q̇σ for the ∇2 Ψ term in Eq. (4.59)); Qu is sourced by Qs and
Qv ≡ τJ QJ /|τ I |, and so on.
For our present purposes the two-field model will suffice. The perturbations then
consist of two scalar degrees of freedom, Qσ and Qs , which obey Eqs. (4.43) and
148
Figure 4-8: The effective mass-squared of the entropy perturbations relative to the Hubble
scale, (µs /H)2 , for the trajectories shown in Fig. 4-2: trajectory 1 (orange dotted line);
trajectory 2 (red solid line); and trajectory 3 (black dashed line). For all three trajectories,
µ2s < 0 while the fields remain near the top of the ridge, since µ2s is related to the curvature
of the potential in the direction orthogonal to the background fields’ evolution. The effective
mass grows much larger than H as soon as the fields roll off the ridge of the potential.
(4.59) (with B I = ΠI = 0) respectively. The effective mass-squared of the entropy
perturbations becomes
µ2s ≡ Mss + 3ω 2 .
(4.67)
If the entropy perturbations are heavy during slow-roll, with µs > 3H/2, then the
amplitude of long-wavelength modes, with k aH, will fall exponentially: Qs ∼
a−3/2 (t) during quasi-de Sitter expansion. For trajectories that begin near the top
of a ridge, on the other hand, the entropy modes will remain light or even tachyonic
at early times, since µ2s is related to the curvature of the potential in the direction
orthogonal to the background fields’ trajectory. Once the background fields roll off
the ridge, the entropy mass immediately grows very large, suppressing further growth
in the amplitude of Qs . See Fig. 4-8.
The perturbations in the adiabatic direction are proportional to the gauge-invariant
curvature perturbation, as derived in Eq. (4.47). Following the usual convention [35],
149
we may define a normalized entropy perturbation as
S≡
H
Qs .
σ̇
(4.68)
In the long-wavelength limit, the coupled perturbations obey general relations of the
form [35]
k2
Ṙc = αHS + O
,
a2 H 2
2 k
,
Ṡ = βHS + O
2
a H2
(4.69)
in terms of which we may write the transfer functions as
Z
t
dt0 α(t0 )H(t0 )TSS (t∗ , t0 ),
t∗
Z t
0
0
0
dt β(t )H(t ) .
TSS (t∗ , t) = exp
TRS (t∗ , t) =
(4.70)
t∗
The transfer functions relate the gauge-invariant perturbations at one time, t∗ , to their
values at some later time, t. We take t∗ to be the time when a fiducial scale of interest
first crossed outside the Hubble radius during inflation, defined by a2 (t∗ )H 2 (t∗ ) = k∗2 .
In the two-field case, both Rc and S are scalars in field space, and hence α, β, TRS ,
and TSS are also scalars. Thus there is no time-ordering ambiguity in the integral for
TSS in Eq. (4.70).
In the two-field case, Eq. (4.51) becomes
Ṙ = 2ωS + O
k2
a2 H 2
.
(4.71)
Comparing with Eq. (4.69), we find
α(t) =
2ω(t)
.
H(t)
(4.72)
The variation of the gauge-invariant curvature perturbation is proportional to the
150
turn-rate. For Ṡ we take the long-wavelength and slow-roll limits of Eq. (4.59):
µ2s
Qs .
3H
(4.73)
µ2s
σ̈
.
−+
2
3H
H σ̇
(4.74)
Q̇s ' −
Eq. (4.69) then yields
β=−
Taking the slow-roll limit of Eq. (4.31) for σ̇, we have
3H σ̇ ' −σ̂ I V,I .
(4.75)
Taking a covariant time derivative of both sides, using the definition of ω I in Eq.
(4.39), and introducing the slow-roll parameter
ηss ≡ Mpl2
Mss
,
V
(4.76)
we arrive at
β = −2 − ηss + ησσ −
4 ω2
,
3 H2
(4.77)
where ησσ is defined in Eq. (4.37). For trajectories that begin near the top of a ridge,
ηss will be negative at early times (like µ2s ), which can yield β > 0. In that case,
TSS (t∗ , t) will grow. If one also has a nonzero turn-rate, ω — and hence, from Eq.
(4.72), a nonzero α within the integrand for TRS (t∗ , t) — then the growing entropy
modes will source the adiabatic mode.
The power spectrum for the gauge-invariant curvature perturbation is defined by
[11, 12]
hRc (k1 )Rc (k2 )i = (2π)3 δ (3) (k1 + k2 )PR (k1 ),
(4.78)
where the angular brackets denote a spatial average and PR (k) = |Rc |2 . The dimensionless power spectrum is then given by
PR (k) =
k3
|Rc |2 ,
2π 2
151
(4.79)
and the spectral index is defined as
ns ≡ 1 +
∂ ln PR
.
∂ ln k
(4.80)
Using the transfer functions, we may relate the power spectrum at time t∗ to its value
at some later time, t, as
2
PR (k) = PR (k∗ ) 1 + TRS
(t∗ , t) ,
(4.81)
where k corresponds to a scale that crossed the Hubble radius at some time t > t∗ .
The scale-dependence of the transfer functions becomes [13, 12, 35, 32],
1 ∂TRS
= −α − βTRS ,
H ∂t∗
1 ∂TSS
= −βTSS ,
H ∂t∗
(4.82)
and hence the spectral index for the power spectrum of the adiabatic fluctuations
becomes
1
ns = ns (t∗ ) +
H
∂TRS
∂t∗
sin (2∆)
(4.83)
where
TRS
.
cos ∆ ≡ p
2
1 + TRS
(4.84)
Given Eq. (4.43) in the limit ωJ δsJ = ωQs 1, the spectral index evaluated at t∗
matches the usual single-field result to lowest order in slow-roll parameters [11, 12, 50]:
ns (t∗ ) = 1 − 6(t∗ ) + 2ησσ (t∗ ).
(4.85)
Scales of cosmological interest first crossed the Hubble radius between 40 and
60 efolds before the end of inflation. In each of the scenarios of Fig. 4-2 the fields
remained near the top of the ridge in the potential until fewer than 40 efolds before
the end of inflation. As indicated in Fig. 4-9, TRS remains small between N∗ = 60 and
40 for each of the three trajectories, with little sourcing of the adiabatic perturbations
152
Figure 4-9: The transfer function TRS for the three trajectories of Fig. 4-2: trajectory
1 (orange dotted line); trajectory 2 (red solid line); and trajectory 3 (black dashed line).
Trajectories 2 and 3, which begin nearer the top of the ridge in the potential than trajectory
1, evolve as essentially single-field models during early times, before the fields roll off the
ridge.
by the entropy perturbations. This behavior of TRS is consistent with the behavior
of ω = αH/2 as shown in Fig. 4-7: ω (and hence α) remains small until the fields
roll off the ridge in the potential. Only in the case of trajectory 1, which began
least high on the ridge among the trajectories and hence fell down the ridge soonest
(at N∗ = 34.5 efolds before the end of inflation), does TRS become appreciable by
N∗ = 40. In particular, we find TRS (N40 ) = 0.530 for trajectory 1; TRS (N40 ) = 0.011
for trajectory 2; and TRS (N40 ) = 0.001 for trajectory 3.
Fixing the fiducial scale k∗ to be that which first crossed the Hubble radius N∗ =
60 efolds before the end of inflation, we find ns (t∗ ) = 0.967 for each of the three
trajectories of Fig. 4-2, in excellent agreement with the observed value ns = 0.971 ±
0.010 [3]. Corrections to ns from the scale-dependence of TRS remain negligible as
long as TRS remains small between N∗ = 60 and 40. Consequently, we find negligible
tilt in the spectral index across the entire observational window for trajectories 2
and 3, whereas the spectral index for trajectory 1 departs appreciably from ns (t∗ ) for
scales that crossed the Hubble radius near N∗ = 40. See Fig. 4-10.
153
Figure 4-10: The spectral index, ns , versus N∗ for the three trajectories of Fig. 4-2:
trajectory 1 (orange dotted line); trajectory 2 (red solid line); and trajectory 3 (black
dashed line). The spectral indices for trajectories 2 and 3 coincide and show no tilt from
the value ns (N60 ) = 0.967.
4.5
Primordial Bispectrum
In the usual calculation of primordial bispectra, one often assumes that the field fluctuations behave as nearly Gaussian around the time t∗ , in which case the three-point
function for the field fluctuations should be negligible. Using the QI construction
of Eq. (4.23), the authors of [36, 37] calculated the action up to third order in perturbations and found several new contributions to the three-point function for field
fluctuations, mediated by the Riemann tensor for the field space, RIJKL . The presence
of the new terms is not surprising; we have seen that RIJKL induces new interactions
among the perturbations even at linear order, by means of the mass-squared matrix,
MIJ in Eq. (4.27). Evaluated at time t∗ , the three-point function for QI calculated
in [37] takes the form
hQI (k1 )QJ (k2 )QK (k3 )i∗ = (2π)3 δ (3) (k1 + k2 + k3 )
×
AIJK
∗
154
+
B∗IJK
+
H∗4
k13 k23 k33
C∗IJK
+
D∗IJK
(4.86)
.
Upon using the definition of σ̂ I in Eq. (4.29), the background equation of Eq. (4.30)
to relate Ḣ to σ̇ 2 , and the definition of in Eq. (4.36), the terms on the righthand
side of Eq. (4.86) may be written [37, 51]
√
AIJK
∗
B∗IJK
C∗IJK
D∗IJK
2 I JK
σ̂ G fA (k1 , k2 , k3 ) + cyclic permutations,
Mpl
√
4Mpl 2
σ̂A RI(JK)A fB (k1 , k2 , k3 ) + cyclic permutations,
=
3
2Mpl2 =
σ̂A σ̂B R(I|AB|J;K) fC (k1 , k2 , k3 ) + cyclic permutations,
3
8Mpl2 =−
σ̂A σ̂B RI(JK)A;B fD (k1 , k2 , k3 ) + cyclic permutations,
3
=
(4.87)
where RIABJ;K = G KM DM RIABJ , and fI (ki ) are shape-functions in Fourier space
that depend on the particular configuration of triangles formed by the wavevectors
ki . Comparable to the findings in [28, 29], each of the contributions to the three-point
function for the field fluctuations is suppressed by a power of the slow-roll parameter,
1.
The quantity of most interest to us is not the three-point function for the field
fluctuations but the bispectrum for the gauge-invariant curvature perturbation, ζ,
which may be parameterized as
hζ(k1 )ζ(k2 )ζ(k3 )i ≡ (2π)3 δ (3) (k1 + k2 + k3 ) Bζ (k1 , k2 , k3 ).
(4.88)
Recall that the two gauge-invariant curvature perturbations, Rc and ζ, coincide in the
long-wavelength limit when working to first order in metric perturbations [11, 12]. In
terms of QI , the δN expansion [53, 54, 56, 55] for ζ on super-Hubble scales becomes
[?]
ζ(xµ ) = (DI N ) QI (xµ ) +
1
(DI DJ N ) QI (xµ )QJ (xµ ) + ...
2
(4.89)
where N = ln |a(tend )H(tend )/k∗ | is the number of efolds after a given scale k∗ first
crossed the Hubble radius until the end of inflation. At t∗ , Eqs. (4.86) and (4.89)
155
yield
hζ(k1 )ζ(k2 )ζ(k3 )i∗ = N,I N,J N,K hQI (k1 )QJ (k2 )QK (k3 )i∗
+
1
(DI DJ N ) N,K N,L
2 Z
d3 q
×
hQI (k1 − q)QK (k2 )i∗ hQJ (q)QL (k3 )i∗ + cyclic perms.
3
(2π)
(4.90)
The bottom two lines on the righthand side give rise to the usual form of fN L , made
suitably covariant to reflect GIJ 6= δIJ . Adopting the conventional normalization, this
term contributes [13, 12, 14, 15, 16]:
hζ(k1 )ζ(k2 )ζ(k3 )ifN L
H∗4
= (2π) δ (k1 + k2 + k3 ) 3 3 3
k1 k2 k3
3
6
,I 2
k1 + k23 + k33
× − fN L N,I N
5
3 (3)
(4.91)
where
fN L = −
5 N ,A N ,B DA DB N
.
6
(N,I N ,I )2
(4.92)
The term on the first line of Eq. (4.90), proportional to the nonzero three-point
function for the field fluctuations, yields new contributions to the bispectrum. However, the three-point function hQI QJ QK i∗ is contracted with the symmetric object,
N,I N,J N,K . Hence we must consider each term within AIJK with care.
In general, the field-space indices, I, J, K, and the momentum-space indices, ki ,
must be permuted as pairs: (I, k1 ), (J, k2 ), (K, k3 ). This is because the combinations arise from contracting the external legs of the various propagators, such as
hQI (k1 )QJ (k2 )i and hQJ (k2 )QK (k3 )i, with the internal legs of each three-point vertex [37, 57]. Let us first consider the special case of an equilateral arrangement in
momentum space, in which k1 = k2 = k3 = k∗ . Then the term proportional to AIJK
156
contributes
H4
hζ(k1 )ζ(k2 )ζ(k3 )iA = (2π)3 δ (3) (k1 + k2 + k3 ) ∗9
4k∗
√
2
(N,I N,J N,K ) σ̂ I G JK + σ̂ J G KI + σ̂ K G IJ fA (k)
×
Mpl
H4
= (2π)3 δ (3) (k1 + k2 + k3 ) ∗9
4k∗
√
3 2 I ×
σ̂ N,I N,A N ,A fA (k),
Mpl
(4.93)
where fA (k) depends only on k. Taking the equilateral limit of the relevant expression
in Eq. (3.17) of [37], we find fA (k) → −5k∗3 /4. Using Eqs. (4.22), (4.29), (4.30), and
(4.36), we also have
σ̂ I N,I =
1 I
1
H
1
√ ,
ϕ̇ DI N = Dt N =
=
σ̇
σ̇
σ̇
Mpl H 2
(4.94)
and hence
"
#
4
3
H
N,A N ,A fA (k).
hζ(k1 )ζ(k2 )ζ(k3 )iA = (2π)3 δ (3) (k1 + k2 + k3 ) ∗9
4k∗ Mpl2
(4.95)
The term arising from B IJK contributes
H4
hζ(k1 )ζ(k2 )ζ(k3 )iB = (2π)3 δ (3) (k1 + k2 + k3 ) ∗9
4k∗
√
4Mpl 2
×
σ̂A N,I N,J N,K RIJKA + RIKJA + cyclic fB (k).
3
(4.96)
But from the symmetry properties of the Riemann tensor we have RIJKA = RKAIJ =
−RAKIJ , and from the first Bianchi identity,
RA[KIJ] = 0.
157
(4.97)
The antisymmetry of the Riemann tensor in its last three indices means that any
contraction of the form
OIJK RAKIJ = 0
(4.98)
for objects OIJK that are symmetric in the indices I, J, K. In our case, we have
OIJK = N,I N,J N,K and thus every term in the square brackets of Eq. (4.96) including
the cyclic permutations may be put in the form of Eq. (4.98). We therefore find
hζ(k1 )ζ(k2 )ζ(k3 )iB = 0
(4.99)
identically in the equilateral limit.
The term arising from C IJK contributes
hζ(k1 )ζ(k2 )ζ(k3 )iC = (2π)3 δ (3) (k1 + k2 + k3 )
H4
4k∗9
2Mpl2 ×
σ̂A σ̂B N,I N,J N,K R(I|AB|J;K) fC (k).
3
(4.100)
In the equilateral limit, we find fC (k∗ ) ' 15k∗3 , based on the limit of the appropriate
expression in Eq. (3.17) of [37]. We may identify the nonzero terms in Eq. (4.100)
using the Bianchi identities. The first Bianchi identity is given in Eq. (4.97), and the
second Bianchi identity may be written
RAB[CD;E] = 0.
(4.101)
Using the (anti)symmetry properties of the Riemann tensor and Eqs. (4.97) and
(4.101), together with the fact that the combinations OIJK ≡ N,I N,J N,K and ΩAB ≡
σ̂A σ̂B are symmetric in their indices, we find the only nonzero term within Eq. (4.100)
to be
hζ(k1 )ζ(k2 )ζ(k3 )iC = (2π)3 δ (3) (k1 + k2 + k3 )
H4
4k∗9
2Mpl2 ×
σ̂A σ̂B N,I N,J N,K RIABJ;K fC (k).
3
158
(4.102)
The final term to consider arises from DIJK . In particular, in the equilaterial limit
we have
hζ(k1 )ζ(k2 )ζ(k3 )iD = (2π)3 δ (3) (k1 + k2 + k3 )
H4
4k∗9
4Mpl2 ×−
σ̂A σ̂B N,I N,J N,K RIJKA;B + RIKJA;B + cyclic fD (k).
3
(4.103)
Again we may use RIJKA = RKAIJ = −RAKIJ and Eq. (4.97) to put the first term
in square brackets in Eq. (4.103) in the form
OIJK RAKIJ;B = 0
(4.104)
for OIJK symmetric. The same occurs for the second term in square brackets in Eq.
(4.103) and for all cyclic permutations of I, J, K. Hence we find
hζ(k1 )ζ(k2 )ζ(k3 )iD = 0
(4.105)
identically in the equilateral limit.
The new nonvanishing terms in Eqs. (4.95) and (4.102) remain considerably
smaller than the fN L term of Eq. (4.91) for the family of models of interest. The
term stemming from AIJK in Eq. (4.95) is proportional to (N,A N ,A ), whereas the
fN L term is multiplied to the square of that term. For models of interest here, in
which the potential includes ridges, the gradient term is significant. For each of the
three trajectories of Fig. 2, for example, (N,A N ,A ) = O(103 ) across the full range
N∗ = 60 to N∗ = 40. The gradient increases as the ratio of ξχ /ξφ increases, and hence
the fN L term will dominate the term coming from AIJK whenever |fN L | > 10−3 .
For the term involving RIABJ;K in Eq. (4.102), we may take advantage of the fact
that for two-field models the Riemann tensor for the field space may be written
RABCD = K(φI ) [GAC GBD − GAD GBC ] ,
159
(4.106)
where K(φI ) is the Gaussian curvature. In two dimensions, K(φI ) = 12 R(φI ), where
R is the Ricci scalar. Since DK GAB = GAB;K = 0 and K(φI ) is a scalar in the field
space, the covariant derivative of the Riemann tensor is simply proportional to the
ordinary (partial) derivative of the Gaussian curvature, K. In particular, we find
σ̂A σ̂B N,I N,J N,K RIABJ;K = − ŝIJ N,I N,J
N,K K,K ,
(4.107)
where ŝIJ ≡ G IJ − σ̂ I σ̂ J is the projection operator for directions orthogonal to the
adiabatic direction. We calculate K in Eq. (4.115). At early times, as the system
undergoes slow-roll inflation, we have ξφ φ2 + ξχ χ2 Mpl2 . For the trajectories as in
Fig. 2, moreover, the system evolves along a ridge such that ξφ φ2 ξχ χ2 . In that
case, we find
K'
1
[1 + 6(ξφ + ξχ ) + 36ξφ (ξχ − ξφ )] ∼ φ0 , χ0 ,
108ξφ2 Mpl2
(4.108)
and hence K,I ∼ 0. Thus, in addition to being suppressed by the slow-roll factor, ,
the contribution to the primordial bispectrum from the RIABJ;K term is negligible in
typical scenarios of interest, because of the weak variation of the Gaussian curvature
of the field-space manifold around the times N∗ = 60 to N∗ = 40 efolds before the end
of inflation. This matches the behavior shown in Fig. 4-5: the field-space manifold is
nearly flat until one reaches the vicinity of φ, χ ∼ 0, near the end of inflation.
Though these results were derived in the equilateral limit, for which k1 = k2 =
k3 = k∗ , we expect the same general pattern to apply more generally, for example,
to the squeezed local configuration in which k1 ' k2 = k∗ and k3 ' 0. As one
departs from the equilateral limit the exact cancellations of Eqs. (4.99) and (4.105)
no longer hold, though each of the components of the field-space Riemann tensor
and its gradients remains small between N∗ = 60 to N∗ = 40 efolds before the end
of inflation for models of the class we have been studying here. Meanwhile, the kdependent functions, fI (ki ) in Eq. (4.87), remain of comparable magnitude to the
k-dependent contribution in Eq. (4.91) [37] — each contributes as [O(1) − O(10)]×k 3
160
— while the coefficients of the additional terms arising from AIJK , B IJK , C IJK , and
DIJK are further suppressed by factors of . For models of the class we have been
studying here, we therefore expect the (covariant version of the) usual fN L term
to dominate the primordial bispectrum. Moreover, given the weak dependence of
the Gaussian curvature K(φI ) with φI between N∗ = 60 and N∗ = 40, we do not
expect any significant contributions to the running of fN L with scale to come from
the curvature of the field-space manifold, given the analysis in [38].
We calculate the magnitude of fN L numerically, following the definition in Eq.
(4.92). The discrete derivative of N along the φ direction is constructed as
N,φ =
N (φ + ∆φ, χ) − N (φ − ∆φ, χ)
,
2∆φ
(4.109)
where N (φ, χ) is the number of efolds between t∗ and tend , where tend is determined
by the physical criterion that ä = 0 (equivalent to = 1). For each quantity, such as
N (φ + ∆φ, χ), we re-solve the exact background equations of motion numerically and
measure how the small variation in field values at t∗ affects the number of efolds of
inflation between t∗ and the time at which ä = 0. The discrete derivatives along the
other field directions and the second derivatives are constructed in a corresponding
manner. Covariant derivatives are calculated using the discrete derivatives defined
here and the field-space Christoffel symbols evaluated at background order. For the
trajectories of interest, the fields violate slow-roll late in their evolution (after they
have fallen off the ridge of the potential), but they remain slowly rolling around the
time t∗ ; if they did not, as we saw in Section IV, then the predictions for the spectral
index, ns (t∗ ) would no longer match observations. We therefore do not consider
separate variations of the field velocities at the time t∗ , since in the vicinity of t∗ they
are related to the field values. Because the second derivatives of N are very sensitive
to the step sizes ∆φ and ∆χ, we work with 32-digit accuracy, for which our numerical
results converge for finite step-sizes in the range ∆φ, ∆χ = {10−6 , 10−5 }.
For the three trajectories of Fig. 4-2, we find the middle case, trajectory 2, yields
a value of fN L of particular interest: |fN L | = 43.3 for fiducial scales k∗ that first
161
Figure 4-11: The non-Gaussianity parameter, |fN L |, for the three trajectories of Fig. 42: trajectory 1 (orange dotted line); trajectory 2 (solid red line); and trajectory 3 (black
dashed line). Changing the fields’ initial conditions by a small amount leads to dramatic
changes in the magnitude of the primordial bispectrum.
crossed the Hubble radius N∗ = 60 efolds before the end of inflation. Note the strong
sensitivity of fN L to the fields’ initial conditions: varying the initial value of χ(τ0 ) by
just |∆χ(τ0 )| = 10−3 changes the fields’ evolution substantially — either causing the
fields to roll off the hill too early (trajectory 1) or not to turn substantially in field
space at all (trajectory 3) — both of which lead to negligible values for fN L . See Fig.
4-11.
4.6
Conclusions
We have demonstrated that multifield models with nonminimal couplings generically
produce the conditions required to generate primordial bispectra of observable magnitudes. Such models satisfy at least three of the four criteria identified in previous
reviews of primordial non-Gaussianities [17, 15], namely, the presence of multiple
fields with noncanonical kinetic terms whose dynamics temporarily violate slow-roll
evolution.
Two distinct features are relevant in this class of models: the conformal stretching
162
of the effective potential in the Einstein frame, which introduces nontrivial curvature
distinct from features in the Jordan-frame potential; and nontrivial curvature of the
induced manifold for the field space in the Einstein frame. So long as the nonminimal
couplings are not precisely equal to each other, the Einstein-frame potential will
include bumps or ridges that will tend to cause neighboring trajectories of the fields
to diverge over the course of inflation. Such features of the potential are generic to
this class of models, and hence are strongly motivated by fundamental physics.
We have found that the curvature of the potential dominates the effects of interest
at early and intermediate stages of inflation, whereas the curvature of the field-space
manifold becomes important near the end of inflation (and hence during preheating).
The generic nature of the ridges in the Einstein-frame potential removes one of the
kinds of fine-tuning that have been emphasized in recent studies of non-Gaussianities
in multifield models, namely, the need to introduce potentials of particular shapes
[16, 29, 31, 32]. (We are presently performing an extensive sweep of parameter space
to investigate how fN L behaves as one varies the couplings ξI , λI , and mI . This
will help determine regions of parameter space consistent with current observations.)
On the other hand, much as in [16, 29, 31, 32], we find a strong sensitivity of the
magnitude of the bispectrum to the fields’ initial conditions. Thus the production
during inflation of bispectra with magnitude |fN L | ∼ O(50) requires fine-tuning of
initial conditions such that the fields begin at or near the top of a ridge in the potential.
A subtle question that deserves further study is whether the formalism and results
derived in this chapter show any dependence on frame. Although we have developed
a formalism that is gauge-invariant with respect to spacetime gauge transformations,
and covariant with respect to the curvature of the field-space manifold, we have
applied the formalism only within the Einstein frame. The authors of [48] recently
demonstrated that gauge-invariant quantities such as the curvature perturbation, ζ,
can behave differently in the Jordan and Einstein frames for multifield models with
nonminimal couplings. The question of possible frame-dependence of the analysis
presented here remains under study. Whether quantities such as fN L show significant
evolution during reheating for this family of models, as has been emphasized for
163
related models [31, 58], likewise remains a subject of further research.
4.7
Acknowledgements
It is a great pleasure to thank Mustafa Amin, Bruce Bassett, Rhys Borchert, Xingang
Chen, Joseph Elliston, Alan Guth, Carter Huffman, Francisco Peña, Katelin Schutz,
and David Seery for helpful discussions. This work was supported in part by the U.S.
Department of Energy (DoE) under contract No. DE-FG02-05ER41360 and in part
by MIT’s Undergraduate Research Opportunities Program.
4.8
4.8.1
Appendix
Field-Space Metric and Related Quantities
Given f (φI ) in Eq. (4.17) for a two-field model, the field-space metric in the Einstein
frame, Eq. (4.6), takes the form
3ξφ2 φ2
Mpl2
1+
,
=
2f
f
2 Mpl
3ξφ ξχ φχ
=
,
2f
f
2 Mpl
3ξχ2 χ2
=
1+
.
2f
f
Gφφ
Gφχ = Gχφ
Gχχ
(4.110)
The components of the inverse metric are
G
φφ
G φχ = G χφ
G χχ
!
2f + 6ξχ2 χ2
,
=
C
!
2f
6ξφ ξχ φχ
=−
,
Mpl2
C
!
2f + 6ξφ2 φ2
2f
=
,
Mpl2
C
2f
Mpl2
164
(4.111)
where we have defined the convenient combination
C(φ, χ) ≡ Mpl2 + ξφ (1 + 6ξφ )φ2 + ξχ (1 + 6ξχ )χ2
(4.112)
= 2f + 6ξφ2 φ2 + 6ξχ2 χ2 .
The Christoffel symbols for our field space take the form
ξφ (1 + 6ξφ )φ ξφ φ
−
,
C
f
ξχ χ
=−
,
2f
ξφ (1 + 6ξχ )φ
=
,
C
ξχ (1 + 6ξφ )χ
,
=
C
ξφ φ
=−
,
2f
ξχ (1 + 6ξχ )χ ξχ χ
−
=
C
f
Γφφφ =
Γφχφ = Γφφχ
Γφχχ
Γχφφ
Γχφχ = Γχχφ
Γχχχ
(4.113)
For two-dimensional manifolds we may always write the Riemann tensor in the
form
RABCD = K(φI ) [GAC GBD − GAD GBC ] ,
(4.114)
where K(φI ) is the Gaussian curvature. In two dimensions, K(φI ) = 21 R(φI ), where
R(φI ) is the Ricci scalar. Given the field-space metric of Eq. (4.110), we find
R(φI ) = 2K(φI ) =
2
2
2
(1
+
6ξ
)(1
+
6ξ
)(4f
)
−
C
.
φ
χ
3Mpl2 C 2
165
(4.115)
166
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field-space,” arXiv:1208.6011 [astro-ph.CO].
[38] C. T. Byrnes and J.-O. Gong, “General formula for the running of fN L ,”
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spectra in inflation,” Phys. Rev. D40, 1753 (1989).
[40] V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, “Theory of cosmological perturbations,” Phys. Rept. 215, 203 (1992).
[41] F. L. Bezrukov and M. E. Shaposhnikov, “The Standard Model Higgs boson as
the inflaton,” Phys. Lett. B659, 703 (2008) [arXiv:0710.3755 [hep-th]].
[42] A. de Simone, M. P. Hertzberg, and F. Wilczek, “Running inflation in the Standard Model,” Phys. Lett. B678 (2009): 1-8 [arXiv:0812.4946 [hep-ph]]; F. L.
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Bezrukov, A. Magnin, and M. E. Shaposhnikov, “Standard Model Higgs boson
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from inflation: two loop analysis,” JHEP 0907 (2009): 089 [arXiv:0904.1537
[hep-ph]]; A. O. Barvinsky, A. Yu. Kamenshchik, C. Kiefer, A. A. Starobinsky, C. Steinwachs, “Asymptotic freedom in inflationary cosmology with a nonminimally coupled Higgs field,” JCAP 0912 (2009): 003 [arXiv:0904.1698 [hepph]].
[43] R. N. Greenwood, D. I. Kaiser, and E. I. Sfakianakis, “Multifield dynamics of
Higgs inflation,” arXiv:1210.8190 [hep-ph].
[44] The leading behavior of Eq. (4.19) in the limit ξφ φ2 Mpl2 is unchanged if we
calculate the curvature of the potential using the covariant derivative for the
field-space, (Dχ2 V )|χ=0 , where DI is defined in Eq. (4.21).
[45] X. Chen, “Primordial features as evidence for inflation,” arXiv:1104.1323 [hepth]; X. Chen, “Fingerprints of primordial universe paradigms as features in
density perturbations,” arXiv:1106.1635 [astro-ph.CO].
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inflation: A possible resolution of the large scale structure problem,” Astrophys. J. 323 (1987): 423-432; D. Polarski and A. A. Starobinsky, “Spectra of
perturbations produced by double inflation with an intermediate matter dominated stage,” Nucl. Phys. B385 (1992): 623; D. Polarski and A. A. Starobinsky,
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model,” Phys. Lett. B356 (1995): 196 [arXiv:astro-ph/9505125].
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[arXiv:1004.3805 [astro-ph.CO]].
172
[48] J. White, M. Minamitsuji, and M. Sasaki, “Cosmological perturbation in multifield inflation with non-minimal coupling,” arXiv:1205.0656 [astro-ph.CO].
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Phys. Rev. D52 (1995): 4295 [arXiv:astro-ph/9408044].
[51] Because the field fluctuations at time t1 live in a different tangent space to
the field-space manifold than do the fluctuations at time t2 , the authors of
[37] introduce propagators and related transport functions to translate between
quantities in the two distinct tangent spaces, building on previous work in [52].
Since the two tangent spaces coincide in the limit t2 → t1 , the propagators
reduce to the identity matrix in that limit. What matters in our discussion is
the symmetry structure of the various terms in Eq. (4.86), which should hold
at any time, hence we evaluate the three-point function at time t∗ .
[52] G. J. Anderson, D. J. Mulryne, and D. Seery, “Transport equations for the
inflationary trispectrum,” arXiv:1205.0024 [astro-ph.CO].
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of the density perturbations produced during inflation,” Prog. Theo. Phys. 95,
71 (1996) [arXiv:astro-ph/9507001].
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the curvature perturbation,” JCAP 0505 (2005): 004 [arXiv:astro-ph/0411220].
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delta N formalism for multi-component inflation,” JCAP 0510 (2005): 004
[arXiv:astro-ph/0506262].
[56] D. H. Lyth and Y. Rodriguez, “The inflationary prediction for primordial nonGaussianity,” Phys. Rev. Lett. 95 (2005): 121302 [arXiv:astro-ph/0504045].
173
[57] We are especially indebted to Joseph Elliston for helpful discussions on this
point.
[58] G. Leung, E. R. M. Tarrant, C. T. Byrnes, and E. J. Copeland, “Reheating,
multifield inflation and the fate of primordial observables,” JCAP 09 (2012):
008 [arXiv:1206.5196 [astro-ph.CO]].
174
Chapter 5
Multifield Dynamics of Higgs
Inflation
Higgs inflation is a simple and elegant model in which early-universe inflation is
driven by the Higgs sector of the Standard Model. The Higgs sector can support
early-universe inflation if it has a large nonminimal coupling to the Ricci spacetime
curvature scalar. At energies relevant to such an inflationary epoch, the Goldstone
modes of the Higgs sector remain in the spectrum in renormalizable gauges, and hence
their effects should be included in the model’s dynamics. We analyze the multifield
dynamics of Higgs inflation and find that the multifield effects damp out rapidly
after the onset of inflation, because of the gauge symmetry among the scalar fields
in this model. Predictions from Higgs inflation for observable quantities, such as the
spectral index of the power spectrum of primordial perturbations, therefore revert to
their familiar single-field form, in excellent agreement with recent measurements. The
methods we develop here may be applied to any multifield model with nonminimal
couplings in which the N fields obey an SO(N ) symmetry in field space.
5.1
Introduction
The recent discovery at CERN of a scalar boson with Higgs-like properties [1] heightens the question of whether the Standard Model Higgs sector could have played inter175
esting roles in the early universe, at energies well above the electroweak symmetrybreaking scale. In particular, the suggestive evidence for the Higgs boson raises the
possibility to return to an original motivation for cosmological inflation, namely, to
realize a phase of early-universe acceleration driven by a scalar field that is part of a
well-motivated model from high-energy particle physics [2, 3, 4].
Higgs inflation [5] represents an elegant approach to building a workable inflationary model based on realistic ingredients from particle physics. In this model, a large
nonminimal coupling of the Standard Model electroweak Higgs sector drives a phase
of early-universe inflation. Such nonminimal couplings are generic: they arise as necessary renormalization counterterms for scalar fields in curved spacetime [6, 7, 8, 9].
Moreover, renormalization-group analyses indicate that for models with matter akin
to the Standard Model, the nonminimal coupling, ξ, should grow without bound
with increasing energy scale [9]. Previous analyses of Higgs inflation have found
that ξ typically grows by at least an order of magnitude between the electroweak
symmetry-breaking scale and the inflationary scale [10, 11, 12].
The Standard Model Higgs sector includes four scalar degrees of freedom: the
(real) Higgs scalar and three Goldstone modes. In renormalizable gauges, all four
scalar fields remain in the spectrum at high energies [13, 14, 12, 16, 17]. Thus the
dynamics of Higgs inflation should be studied as a multifield model with nonminimal
couplings. An important feature of multifield models, which is absent in singlefield models, is that the fields’ trajectories can turn within field space as the system
evolves. Such turns are a necessary (but not sufficient) condition for multifield models
to depart from the empirical predictions of simple single-field models [18, 19, 20, 21,
22, 23, 24].
In this chapter we analyze the background dynamics of Higgs inflation, in which all
four scalar fields of the Standard Model electroweak Higgs sector have nonminimal
couplings. We find that multifield dynamics damp out quickly after the onset of
inflation, before perturbations on cosmologically relevant length scales first cross the
Hubble radius. As regards observable quantities like the power spectrum of primordial
perturbations, the model therefore behaves effectively as a single-field model. The
176
multifield dynamics remain subdominant in Higgs inflation because of the particular
symmetries of the Higgs sector. Closely related models, which lack those symmetries,
can produce conspicuous departures from the single-field case [24].
We are principally interested here in the behavior of classical background fields
and long-wavelength perturbations, which behave essentially classically. Therefore we
bracket, for this analysis, the question of the unitarity of Higgs inflation. Conflicting
conclusions have been advanced regarding whether the appropriate renormalization
√
cut-off scale for this model should be Mpl , Mpl / ξ, or Mpl /ξ, where Mpl ≡ (8πG)−1/2
is the reduced Planck mass [14, 15, 12, 16, 25]. Even if Higgs inflation might conclusively be shown to violate unitarity, the techniques developed here for the analysis
of multifield dynamics will be relevant for related models that incorporate multiple
scalar fields with nonminimal couplings and symmetries (such as gauge symmetries)
that enforce specific relations among the couplings of the model. In particular, we
expect that multifield effects in models with N scalar fields, in which the scalar fields
obey an SO(N ) symmetry, should damp out rapidly.
In Section II, we briefly introduce the multifield formalism and establish notation.
We apply the formalism to Higgs inflation in Section III, and in Section IV we analyze
the behavior of the turn-rate, which quantifies the rate at which the background
trajectory of the system deviates from a single-field case. We study how quickly
the turn-rate damps to zero, both analytically and numerically, confirming that for
Higgs inflation the turn-rate becomes negligible within a few efolds after the start
of inflation. In Section V we turn to implications for observable features of the
primordial power spectrum, confirming that multifield Higgs inflation reproduces the
empirical predictions of previous single-field studies. Concluding remarks follow in
Section VI.
5.2
Multifield Dynamics
Following the approach established in [24], we consider models with N scalar fields
in (3 + 1) spacetime dimensions. We use Greek letters to label spacetime indices,
177
µ, ν = 0, 1, 2, 3; lower-case Latin letters to label spatial indices, i, j = 1, 2, 3; and
upper-case Latin letters to label field-space indices, I, J = 1, 2, ..., N . We also work
in terms of the reduced Planck mass, Mpl ≡ (8πG)−1/2 . In the Jordan frame, the
action takes the form
p
1
µν
I
J
I
I
d x −g̃ f (φ )R̃ − δIJ g̃ ∂µ φ ∂ν φ − Ṽ (φ ) .
2
Z
4
SJordan =
(5.1)
Here f (φI ) is the nonminimal coupling function, and we use tildes for quantities in
the Jordan frame. We perform a conformal transformation to the Einstein frame by
rescaling the spacetime metric tensor,
gµν (x) =
2
f (φI (x)) g̃µν (x),
Mpl2
(5.2)
so that the action in the Einstein frame becomes [26]
Z
SEinstein =
2
Mpl
√
1
µν
I
J
I
d x −g
R − GIJ g ∂µ φ ∂ν φ − V (φ ) .
2
2
4
(5.3)
The potential in the Einstein frame, V , is related to the Jordan-frame potential, Ṽ ,
as
Mpl4
V (φ ) = 2 I Ṽ (φI ),
4f (φ )
I
(5.4)
and the coefficients of the noncanonical kinetic terms are [27, 26]
Mpl2
3
GIJ (φ ) =
δIJ +
f,I f,J ,
2f (φI )
f (φI )
K
(5.5)
where f,I = ∂f /∂φI . The nonminimal couplings induce a field-space manifold in the
Einstein frame that is not conformal to flat; GIJ serves as a metric on the curved
manifold [26]. Therefore we adopt the covariant approach of [24], which respects the
curvature of the field-space manifold.
Varying Eq. (5.3) with respect to φI yields the equation of motion,
φI + g µν ΓIJK ∂µ φJ ∂ν φK − G IK V,K = 0,
178
(5.6)
where φI ≡ g µν φI;µ;ν and ΓIJK (φL ) is the Christoffel symbold for the field-space
manifold, calculated in terms of GIJ . We expand each scalar field to first order
around its classical background value,
φI (xµ ) = ϕI (t) + δφI (xµ ),
(5.7)
and also expand the scalar degrees of freedom of the spacetime metric to first order
around a spatially flat Friedmann-Robertson-Walker (FRW) metric [28, 29, 30]
ds2 = −(1 + 2A)dt2 + 2a(∂i B)dxi dt + a2 [(1 − 2ψ)δij + 2∂i ∂j E] dxi dxj ,
(5.8)
where a(t) is the scale factor. We further introduce a covariant derivative with respect
to the field-space metric and a directional derivative along the background fields’
trajectory, such that for any vector AI in the field-space manifold we have
DJ AI = ∂J AI + ΓIJK AK ,
(5.9)
Dt AI ≡ ϕ̇J DJ AI = ȦI + ΓIJK AJ ϕ̇K ,
where overdots denote derivatives with respect to cosmic time, t.
To background order, Eq. (5.6) becomes
Dt ϕ̇I + 3H ϕ̇I + G IK V,K = 0,
(5.10)
where all quantities involving GIJ (φK ), V (φI ), and their derivatives are evaluated at
background order in the fields: GIJ → GIJ (ϕK ) and V → V (ϕI ). Following [18] we
distinguish between adiabatic and entropic directions in field space by introducing a
unit vector
σ̂ I ≡
ϕ̇I
,
σ̇
(5.11)
where
σ̇ ≡ |ϕ̇I | =
p
GIJ ϕ̇I ϕ̇J .
179
(5.12)
The operator
ŝIJ ≡ G IJ − σ̂ I σ̂ J
(5.13)
projects onto the subspace orthogonal to σ̂ I . Eq. (5.10) then simplifies to
σ̈ + 3H σ̇ + V,σ = 0
(5.14)
V,σ ≡ σ̂ I V,I .
(5.15)
where
The background dynamics likewise take the simple form
1
1 2
H =
σ̇ + V ,
3Mpl2 2
1 2
Ḣ = −
σ̇ ,
2Mpl2
2
(5.16)
where H ≡ ȧ/a is the Hubble parameter.
We may also separate the perturbations into adiabatic and entropic directions.
Working to first order in perturbations, we introduce the gauge-invariant MukhanovSasaki variables [28, 29, 30, 31]
QI ≡ δφI +
ϕ̇I
ψ
H
(5.17)
and the projections
Qσ ≡ σ̂I QI ,
I
I
(5.18)
J
δs ≡ ŝ J Q .
The gauge-invariant curvature perturbation may be defined as Rc ≡ ψ−[H/(ρ+p)]δq,
where the perturbed energy-momentum flux is given by T 0i = ∂i δq [29, 30]. We then
find that Rc is proportional to Qσ [24]
Rc =
H
Qσ .
σ̇
180
(5.19)
Expanding Eq. (5.6) to first order and using the projections of Eq. (5.18), the
perturbations Qσ and δsI obey [24]
"
#
a3 σ̇ 2
Qσ
H
!
V,σ Ḣ
+
ωJ δsJ
σ̇
H
k2
1 d
Q̈σ + 3H Q̇σ + 2 + Mσσ − ω 2 − 2 3
a
Mpl a dt
d
ωJ δsJ − 2
=2
dt
(5.20)
and
Dt2 δsI
2
k I
σ̈
I
I
I
I
I
+ 3Hδ J + 2σ̂ ωJ Dt δs + 2 δ J + M J − 2σ̂ MσJ + ωJ
δsJ
a
σ̇
"
#
(5.21)
Ḣ
σ̈
I
= −2ω Q̇σ + Qσ − Qσ ,
H
σ̇
where the mass-squared matrix is
MIJ ≡ G IK (DJ DK V ) − RILM J ϕ̇L ϕ̇M ,
(5.22)
MσJ ≡ σ̂I MIJ , Mσσ ≡ σ̂I σ̂ J MIJ .
The turn-rate [23, 24] is given by
1
ω I ≡ Dt σ̂ I = − V,K ŝIK ,
σ̇
(5.23)
and ω ≡ |ω I |. Eqs. (5.20) and (5.21) decouple if the turn-rate vanishes, ω I = 0. In
that case, Qσ evolves just as in the single-field case [28, 29, 30, 23, 24]. Given Eq.
(5.19), that means that the power spectrum of primordial perturbations, PR , would
also evolve as in single-field models. Thus a necessary (but not sufficient) condition
for multifield models of this form to deviate from the empirical predictions of simple
single-field models is for the turn-rate to be nonnegligible for some duration of the
fields’ evolution, ω I 6= 0.
181
5.3
Application to Higgs Inflation
The matter contribution to Higgs inflation [5] consists of the Standard Model electroweak Higgs sector, which may be written as a doublet of complex scalar fields,

h=
+
h
0
h

.
(5.24)
The complex fields h+ and h0 may be further decomposed into (real) scalar degrees
of freedom,
1
h+ = √ χ1 + iχ2 ,
2
1
h0 = √ φ + iχ3 ,
2
(5.25)
where φ is the Higgs scalar and χa (with a = 1, 2, 3) are the Goldstone modes. In the
Jordan frame, the potential Ṽ (φI ) depends only on the combination
h† h =
1 2
φ + χ2 ,
2
(5.26)
where χ = (χ1 , χ2 , χ3 ) is a 3-vector of the Goldstone fields. In particular, the
symmetry-breaking potential may be written
Ṽ (φI ) =
2
λ 2
φ + χ2 − v 2 ,
4
(5.27)
in terms of the vacuum expectation value, v. For the Standard Model, v = 246 GeV
Mpl . For Higgs inflation, the nonminimal coupling function is given by
f (φI ) =
1 2
M02
+ ξh† h =
M0 + ξ φ2 + χ2 ,
2
2
(5.28)
where M02 ≡ Mpl2 − ξv 2 and ξ > 0 is the dimensionless nonminimal coupling constant.
In Higgs inflation, we take ξ ∼ O(104 ) [5], and therefore we may safely set M02 =
Mpl2 . In the Einstein frame, the potential gets stretched by the nonminimcal coupling
182
function f (φI ) according to Eq. (5.4). Given Eqs. (5.27) and (5.28), this yields
2
λMpl4 (φ2 + χ2 − v 2 )
V (φ ) = 2 .
4 Mpl2 + ξ (φ2 + χ2 )
I
(5.29)
The model is thus symmetric under rotations among φ and χa that preserve the magp
nitude φ2 + χ2 . When written in the “Cartesian” field-space basis of Eq. (5.25), in
other words, the SU (2) electroweak gauge symmetry manifests as an SO(4) spherical
symmetry in field space.
For any model with N real-valued scalar fields that respects an SO(N ) symmetry, the background dynamics depend on just three initial conditions: the initial
magnitude and initial velocity along the radial direction in field space, and the initial
velocity perpendicular to the radial direction. Without loss of generality, therefore,
we may analyze the background dynamics of Higgs inflation in terms of just two
real-valued scalar fields, φ and χ, and we may set χ(0) = 0, specifying only initial
values for φ(0), φ̇(0), and χ̇(0). This reduction in the effective number of degrees of
freedom stems entirely from the gauge symmetry of the Standard Model electroweak
sector. The remaining dependence on χ̇, meanwhile, distinguishes the background
dynamics from a genuinely single-field model, in which one neglects the Goldstone
fields altogether. For the remainder of this chapter, we exploit the gauge symmetry
to consider only a single Goldstone mode, χ → χ, reducing the problem to that of a
two-field model. Then f (φI ) and V (φI ) depend on the background fields only in the
combination
r≡
p
φ2 + χ2 .
(5.30)
Previous analyses [5, 27, 32, 33, 34] which considered single-field versions of this
model (neglecting the Goldstone modes) found successful inflation for field values
ξφ2 Mpl2 . We confirm this below for the multifield case including the Goldstone
modes. The reason is easy to see from Eq. (5.29). In the limit ξ(φ2 +χ2 ) = ξr2 Mpl2 ,
183
Figure 5-1: The potential for Higgs inflation in the Einstein frame, V (φ, χ). Note the
flattening of the potential for large field values, which is quite distinct from the behavior of
the Jordan-frame potential, Ṽ (φ, χ) in Eq. (5.27).
the potential in the Einstein frame becomes very flat, approaching
2 Mpl
λMpl4
1
+
O
.
V (φ ) →
4ξ 2
ξr2
I
(5.31)
See Fig. 5-1.
Given ξ ∼ 104 , the initial energy density for this model lies well below the Planck
scale, ρ ' V ' λMpl4 /ξ 2 ∼ 10−9 Mpl4 . In fact, as we will see, successful slow-roll inflation (producing at least 70 efolds of inflation) occurs for initial values of the fields
below the Planck scale, unlike in models of chaotic inflation with polynomial potentials that lack nonminimal couplings. Moreover, as emphasized in [5], the flattening
of the potential in the Einstein frame at large field values makes Higgs inflation easily
compatible with the latest observations of the spectral index, ns . Ordinary chaotic
inflation with a λφ4 potential and minimal coupling, on the other hand, yields a
spectral index outside the 95% confidence interval for the best-fit value of ns [35, 36].
Below we confirm this behavior for Higgs inflation even when the Goldstone degrees
of freedom are included.
The field-space metric GIJ is determined by the nonminimal coupling function,
f (φI ), and its derivatives. Explicit expressions for the components of GIJ for a two184
field model with arbitrary couplings, ξφ and ξχ , are given in the Appendix of [24]. In
the case of Higgs inflation, the SU (2) gauge symmetry enforces ξφ = ξχ = ξ. Given
this symmetry, the convenient combination, C(φI ), introduced in the Appendix of
[24] becomes
C(φI ) = 2f + 6ξ 2 φ2 + χ2 = Mpl2 + ξ(1 + 6ξ)r2 .
(5.32)
For ξφ = ξχ = ξ, the Ricci curvature scalar for the field-space manifold, as calculated
in [24], takes the form
R=
4ξ 2
C
+
3ξM
pl .
C2
(5.33)
During inflation, when ξr2 Mpl2 , this reduces to
R→
2
Mpl−2 ,
2
3ξr
(5.34)
indicating that the field-space manifold has a spherical symmetry with radius of
√
curvature rc ∼ ξ r. As shown in [24], the curvature of the field-space manifold
remains negligible in such models until the fields satisfy ξr2 Mpl2 , near the end of
inflation.
From Eq. (5.10), and using the expressions for G IJ and ΓIJK in the Appendix of
[24], the equation of motion for the background field φ(t) takes the form
f˙
2
2
ξ(1 + 6ξ) 2
4 φ(φ + χ )
2
φ̈ + 3H φ̇ +
φ φ̇ + χ̇ − φ̇ + λMpl
= 0.
C
f
2f C
(5.35)
The equation for χ follows upon replacing φ ←→ χ. Using Eq. (5.12), the square of
the fields’ velocity vector becomes
σ̇ 2 =
Mpl2
2f
"
φ̇2 + χ̇2
#
3f˙2
+
,
f
(5.36)
and the gradient of the potential in the direction σ̂ I becomes
σ̂ I V,I = V,σ =
λMpl6 (φ2 + χ2 ) f˙
.
ξ(2f )3
σ̇
185
(5.37)
We may verify that multifield Higgs inflation exhibits slow-roll behavior for typical
choices of couplings and initial conditions. First consider the single-field case, in which
we set χ = χ̇ = 0. Near the start of inflation (with ξφ2 Mpl2 ), the terms in Eq.
(5.35) that stem from the field’s noncanonical kinetic term take the form
ξ(1 + 6ξ) 2 f˙
φ̇2
φφ̇ − φ̇ → − .
C
f
φ
(5.38)
The usual slow-roll requirement for single-field models, |φ̇| |Hφ|, ensures that the
terms in Eq. (5.38) remain much less than the 3H φ̇ term in Eq. (5.35). Neglecting
φ̈, the single-field, slow-roll limit of Eq. (5.35) becomes
λMpl4
,
6ξ 3 φ
(5.39)
√
λ Mpl3
φ̇ ' − √
.
3 3 ξ2φ
(5.40)
3H φ̇ ' −
or, upon using H 2 ' V /(3Mpl2 ),
Setting ξ = 104 and fixing the initial field velocity by Eq. (5.40) requires φ(0) ≥
0.1Mpl to yield N ≥ 70 efolds of inflation in the single-field case.
A much broader range of initial conditions yields N ≥ 70 efolds in the two-field
case. From Eq. (5.16) we see that inflation (with ä > 0) requires σ̇ 2 V . Given the
SO(N ) symmetry of the model, we may set χ(0) = 0 without loss of generality, and
parameterize the fields’ initial velocities as
√
λ Mpl3
x,
φ̇(0) = √
3 3 ξ 2 φ(0)
√
λ Mpl3
χ̇(0) = √
y
3 3 ξ 2 φ(0)
(5.41)
in terms of dimensionless constants x and y. (The single-field case corresponds to
x = −1, y = 0.) Near the start of inflation, when ξr2 = ξφ2 Mpl2 , Eq. (5.36)
186
becomes
2
σ̇ |χ(0)=0 →
λMpl4
4ξ 2
Mpl2
ξφ2 (0)
2
4 (1 + 6ξ)x2 + y 2 .
27ξ
(5.42)
The first term in parentheses is just the value of the potential, V , near the start
of inflation, as given in Eq. (5.31). The second term in parentheses is small near
the beginning of inflation, given ξr2 Mpl2 . Hence the initial values for φ̇ and χ̇,
parameterized by the coefficients x and y, may be substantially larger than in the
single-field case while still keeping σ̇ 2 V .
Fig. 5-2 shows H(t), φ(t), and χ(t) for a scenario in which φ̇(0) and χ̇(0) greatly
exceed the single-field relation of Eq. (5.40): |x| = 102 and |y| = 106 . As is evident
in the figure, the large initial velocities cause the fields to oscillate rapidly. The extra
kinetic energy makes the initial value of H(t) larger than in the corresponding singlefield case. The increase in H, in turn, causes the fields’ velocities to damp out even
more quickly, due to the 3H φ̇ and 3H χ̇ Hubble-drag terms in each field’s equation of
motion. Thus the system rapidly settles into a slow-roll regime that continues for 70
efolds. As shown in Fig. 3, we may achieve N ≥ 70 efolds with even smaller initial
field values by making the initial field velocities correspondingly larger.
5.4
Turn Rate
The components of the turn-rate, ω I in Eq. (5.23), take the form
λMpl4 r2
ωφ = −
σ̇ 2f
"
#
Mpl2 φ̇ φ
− 2 2 φφ̇ + χχ̇ .
C
4f σ̇
(5.43)
The other component, ω χ , follows upon replacing φ ←→ χ. The length of the turnrate vector is given by
q
p
1
I
J
ω = |ω | = GIJ ω ω =
ŝKM V,K V,M ,
σ̇
I
187
(5.44)
Figure 5-2: The evolution of H(t) (black dashed line) and the fields φ(t) (red solid line) and
χ(t) (blue dotted line).
√ The fields are measured in3 units of Mpl and we use the dimensionless
time variable τ = λ Mpl t. We have plotted 10 H so that its scale is commensurate with
the magnitude of the fields. The Hubble parameter begins large, H(0) = 8.1 × 10−4 , but
quickly falls by a factor of 30 as it settles to its slow-roll value of H = 2.8 × 10−5 . Inflation
proceeds for ∆τ = 2.5 × 106 to yield N = 70.7 efolds of inflation. The solutions shown here
correspond to ξ = 104 , φ(0) = 0.1, χ(0) = 0, φ̇(0) = −2 × 10−6 , and χ̇(0) = 2 × 10−2 . For
the same value of φ(0), Eq. (5.40) corresponds to φ̇(0) = −2 × 10−8 for the single-field case.
where the final expression follows upon using the definition of ω I in Eq. (5.23) and
the identity ŝKM = ŝKA ŝM A , which follows from Eq. (5.13). We find
σ̇ 2 ω 2 = ŝKM V,K V,M =
λ2 Mpl10 6 r C − ξ 2 r2 − (V,σ )2 .
5
(2f ) C
(5.45)
The evolution of the turn rate for typical initial conditions is shown in Fig. 5-4.
In order to analyze the evolution of the background fields, it is easier to move
from Cartesian to polar coordinates, in which the angular velocity and turn rate have
more intuitive behavior. In addition to the radius, r2 = φ2 + χ2 , we also define the
angle
χ
γ ≡ arctan
.
φ
(5.46)
Single-field trajectories correspond to constant γ(t). In the polar coordinate system,
188
Figure 5-3: Contour plots showing the number of efolds of inflation as one varies the fields’
initial conditions, keeping ξ = 104 fixed. In each panel, the vertical axis is χ̇(0) and the
horizontal axis is φ̇(0). The panels correspond to φ(0) = 10−1 Mpl (top left), 10−2 Mpl
(top right), 5 × 10−3 M√pl (bottom left) and 10−4 Mpl (bottom right), and we again use
dimensionless time τ = λ Mpl t. In each panel, the line for N = 70 efolds is shown in bold.
Note how large these initial velocities are compared to the single-field expectation of Eq.
(5.40).
189
Figure 5-4: Evolution of the turn rate. The left picture shows the evolution with initial
conditions as in Fig. 5-2. The right figure has √
initial conditions φ(0) = 0.1, χ(0) = φ̇(0) = 0,
and χ̇(0) = 2 × 10−5 in units of Mpl and τ = λ Mpl t. In both cases we set ξ = 104 . Recall
from Fig. 5-2 that inflation lasts until τend ∼ O(106 ) for these initial conditions; hence we
find that ω damps out within a few efolds after the start of inflation.
the background dynamics of Eq. (5.16) may be written
"
#
2
2
4
λM
1
3ξ
r
pl
r2 ṙ2 +
H2 =
ṙ2 + r2 γ̇ 2 +
,
12f
f
2 (Mpl2 + ξr2 )
1
3ξ 2 2 2
2
2 2
Ḣ = −
ṙ + r γ̇ +
r ṙ .
4f
f
(5.47)
The equations of motion become
r̈ + 3H ṙ − rγ̇ 2 +
ξ
ξ(1 + 6ξ)
r3
r ṙ2 + r2 γ̇ 2 − ṙ2 r + λMpl4
=0
C
f
2f C
and
γ̈ +
Mpl2
ṙ
3H + 2
r (Mpl2 + ξr2 )
(5.48)
!
γ̇ = 0.
(5.49)
In this new basis the turn rate may be written compactly as
λ2 Mpl8
ω2 =
2f C
r4 γ̇
r2 γ̇ 2 (Mpl2 + ξr2 ) + ṙ2 C
!2
(5.50)
This expression vanishes in both the limits |γ̇| → 0 and |γ̇| → ∞: if the angular
velocity is either too large or too small, the fields’ evolution reverts to effectively
190
single-field behavior (either purely radial motion or purely angular motion). Of the
two limits, however, only pure-radial motion is stable. It is ultimately the evolution
of γ(t) that will determine the fate of the turn rate.
It is obvious from Eq. (5.49) that the line γ̇ = 0 is the fixed point of the angular
motion. The character of the fixed point is defined by the sign of the γ̇ term, which
is less trivial. It can be negative close to r = 0 due to the high curvature of the field
manifold and the small value of the Hubble parameter, but in the slow-roll regime of
the radial field, with ξr2 Mpl2 , the sign of γ̇ is safely positive. That means that we
can treat the angular motion as damped throughout inflation.
For large nonminimal coupling and/or slow rolling of the radial field the last term
in Eq. (5.49) may be neglected, which yields
γ̈ + 3H γ̇ = 0.
(5.51)
The only complicated object in Eq. (5.51) is the Hubble parameter, which may be
simplified in the limit of a slow rolling radial field and large nonminimal coupling
upon making use of Eq. (5.47):
1
H'√
6ξ
s
γ̇ 2 +
λMpl2
.
2ξ
(5.52)
Then Eq. (5.51) becomes
3
γ̈ + √
6ξ
s
γ̇ 2
λMpl2
+
γ̇ ' 0.
2ξ
(5.53)
Although Eq. (5.53) can be solved exactly (see the Appendix), it is instructive to
examine the two limits of large and small γ̇, which provide most of the relevant
information.
For small angular velocity, γ̇ q
λMpl2 /2ξ, we recover the linear limit
s
γ̈ +
3
ξ
λMpl2
γ̇ ' 0
12
191
(5.54)
with the solution
" √
#
3λ
Mpl t ∝ e−3N ,
γ̇ = γ̇0 exp −
2ξ
(5.55)
where N = Ht. It is very easy to measure time in efolds in this limit, since the
Hubble parameter is nearly constant. Eq. (5.55) illustrates that any small, initial
angular velocity will be suppressed within a couple of efolds, or equivalently within a
√
time of the order of ξ/( λ Mpl ).
In the opposite limit, γ̇ q
λMpl2 /2ξ, which we call the nonlinear regime, Eq.
(5.53) becomes
1
γ̈ + 3 √ γ̇ 2 ' 0
6ξ
with the solution
1
3t
γ̇ =
+√
γ̇0
6ξ
(5.56)
−1
.
(5.57)
Given Eqs. (5.55) and (5.57), we may follow the evolution of any initial angular
velocity. If γ̇ begins large enough it will start in the nonlinear regime, where it will
q
stay until it becomes of order λMpl2 /2ξ. We parameterize the cross-over regime as
γ̇ =
√
z
λMpl √
2ξ
(5.58)
where z is some constant of order one. The cross-over time may then be estimated
by inverting Eq. (5.57) to find
#
√ "r
6ξ
2ξ 1
1
tnl =
−
.
3
λ Mpl z γ̇0
(5.59)
There exists an upper limit on the time it takes for the angular velocity to decay,
namely
√
2 ξ
λMpl tnl,max = √ .
3z
(5.60)
We have verified all of these analytic predictions using numerical calculations of
the exact equations for the coupled two-field system in an expanding universe. In
Fig. 5-5 we plot the number of efolds from the beginning of inflation at which the
192
turn rate reaches its maximum value, as we vary the fields’ initial velocities. Note
that for any combination of initial conditions that yields at least Ntot = 70 efolds, ω
reaches its maximum value between N (ωmax ) = 3.5 and 5 efolds from the start of the
fields’ evolution (for the range of initial conditions considered there). In Fig. 5-6 we
plot ω as a function of time as we vary the initial angular velocity, γ̇(0). The curves
in red correspond to initial conditions in the linear regime, while the curves in blue
start in the nonlinear regime. Note that the curves starting in the nonlinear regime
have the same amplitude. The existence of a maximum time, tnl,max , is evident from
√
the bunching of the blue curves. We find λMpl tnl,max = τnl,max ∼ few × ξ ∼ 104 , as
expected from Eq. (5.60). In these units and for the initial conditions used in Fig.
5-6, inflation lasts until τend ∼ O(106 ), so τnl,max occurs very early after the onset of
inflation.
Eq. (5.55) shows that the linear region lasts at most a few efolds, so the duration
of the nonlinear region is what will ultimately determine whether or not multifield
effects will persist until observationally relevant length scales first cross the Hubble
√
radius. In the nonlinear regime, Eq. (5.52) yields H ' γ̇/ 6ξ with γ̇ given by Eq.
(5.57). The number of efolds for which the nonlinear regime persists is given by
Z
Nnl =
0
tnl
1
Hdt ' √
6ξ
tnl
Z
0
1
γ̇dt = ln
3
r
2ξ γ̇0
λ Mpl z
!
.
(5.61)
We examine Eq. (5.61) numerically by fixing ξ = 104 and φ(0) = 0.1Mpl and choosing
pairs of initial velocities, φ̇(0) and χ̇(0), that yield 70 efolds (see Fig. 5-7, left); and
also by setting φ̇(0) to various constant values and varying χ̇(0) (Fig. 5-7, right). The
results fall neatly along a least-squares logarithmic fit, as expected from Eq. (5.61).
The function Nnl grows slowly. In order for multifield effects to remain important
more than a few efolds after the start of inflation, the initial angular velocity would
need to be enormous: at least ten orders of magnitude larger than typical values of
the initial field velocity for single-field inflation, as given in Eq. (5.40). We do not
know of any realistic mechanism that could generate initial field velocities so large.
Moreover, for many combinations of initial conditions shown in the righthand side of
193
Figure 5-5: Contour plots showing the number of efolds at which the maximum of the
turn rate occurs, as one varies the fields’ initial conditions. In each panel, the vertical axis
is χ̇(0) and the horizontal axis is φ̇(0). The panels correspond to φ(0) = 10−1 Mpl (top
left), 10−2 Mpl (top right), 5 × 10−3 Mpl (bottom left)√and 10−4 Mpl (bottom right). We set
ξ = 104 and use the dimensionless time-variable τ = λ Mpl t. The thick black curve is the
contour line of initial conditions that yield N = 70 efolds.
194
Figure 5-6: The turn rate as a function of time for different values of the initial angular
0.01
velocity. The parameters used are ξ = 104 , φ(0) = 0.1Mpl , φ̇(0) = χ(0) = 0, and √
≤
2ξ
√
100
γ̇(0) ≤ √2ξ , in terms of dimensionless time, τ = λ Mpl t. In these units and for φ(0) =
0.1Mpl , inflation lasts until τend ∼ O(106 ).
Fig. 5-7, Ntot > 70 efolds (several sets of initial conditions yield Ntot ∼ 90 efolds). For
those scenarios, the turn rate reaches its maximum value deep within the early phase
of the system’s evolution, long before observationally relevant perturbations first cross
the Hubble radius. The multifield dynamics for this model thus behave similarly to
those in related multifield models of inflation that involve the Higgs sector, such as
[37].
We may consider the behavior of a(t) and H(t) in the two different regimes more
closely. From the definition of H and Ḣ in Eq. (5.47) and neglecting the terms
proportional to ṙ (which is equivalent to requiring the field to be slow rolling along
the radial direction), we find
ä
1
= Ḣ + H 2 =
a
12f
λMpl2
r4
2 2
−2r γ̇ +
2 Mpl2 + ξr2
!
.
(5.62)
When the potential dominates we recover what we called the linear regime in the
analysis of the decay of ω. In that regime
ä
> 0,
a
(5.63)
which is an accelerated expansion or cosmological inflation. However, in the nonlinear
195
Figure 5-7: Number of efolds until the maximum value of the turn rate is reached, as a
function of χ̇(0). On the left we plot N (ωmax ) for initial conditions that yield Ntot = 70
efolds; on the right we plot the same quantity for various values of φ̇(0). The logarithmic
fit is an excellent match to our analytic result, Eq. (5.61).
regime, when γ̇ dominates, the situation reverses and we find
ä
1
= − r2 γ̇ 2 < 0,
a
6f
(5.64)
which is an expansion and a very rapid one (because of the large value of H), but it is
not inflation. Regardless of whether we have true inflation or simply rapid expansion
at early times, we may always define the number of efolds as
Z
tend
Hdt.
N=
(5.65)
tin
Thus we may use N as our clock and measure time in efolds from the beginning of the
system’s evolution, regardless of whether it is in the inflationary phase or not. The
fact that in the nonlinear regime the universe is not inflating only makes our results
stronger: all multifield effects decay before the observable scales exit the horizon in a
model that produces enough inflation to solve the standard cosmological problems.
As a final test of our analysis we set ξ = 102 instead of ξ = 104 . The smaller value
of the nonminimal coupling does not lead to a viable model of Higgs inflation — the
WMAP normalization of the power spectrum requires a larger value of ξ [5] — but
we may nonetheless study the dynamics of such a model. We collect the important
196
information about the dynamics of this model in Fig. 5-8. As expected, the model
can provide 70 or more efolds of inflation for a wide range of parameters, and the
corresponding turn rate peaks well before observationally relevant length scales first
crossed the Hubble radius, even when we increase χ̇(0) to a few hundred in units
√
of τ = λMpl t. The excellent logarithmic fit of the time at which the turn rate is
maximum versus χ̇(0) is again evident. Finally the curves of the turn rate versus time
show the same qualitative and quantitative characteristics as Fig. 5-6 for ξ = 104 .
Specifically, if one rescales time and the turn rate appropriately by ξ, the two sets of
curves would be hardly distinguishable.
5.5
Implications for the Primordial Spectrum
We have found that in models with an SO(N ) symmetry among the scalar fields, the
turn rate quickly damps to negligible magnitude within a few efolds after the start of
inflation. In this section we confirm that such behavior yields empirical predictions
for observable quantities like the primordial power spectrum of perturbations that
reproduce expectations from corresponding single-field models.
For models that behave effectively as two-field models, which includes the class of
SO(N )-symmetric models we investigate here, we may distinguish two scalar perturbations: the perturbations in the adiabatic direction, Qσ defined in Eq. (5.18), and
a scalar entropic perturbation [24],
Qs ≡
ωI I
δs .
ω
(5.66)
We noted in Eq. (5.19) that Qσ is proportional to the gauge-invariant curvature
perturbation, Rc . We adopt a similar normalization for the entropy perturbation,
S≡
H
Qs .
σ̇
197
(5.67)
Figure 5-8: Dynamics of our two-field model with ξ = 102 , φ(0) = 1Mpl , and χ(0) = 0.
Clockwise from top left: (1) Contour plot showing the number of efolds as one varies the
fields’ initial conditions. The thick curve corresponds to 70 efolds. (2) Contour plot showing
the number of efolds at which the maximum of the turn rate occurs, as one varies the fields’
initial conditions. The thick curve corresponds to Ntot = 70 efolds. (3) Number of efolds
until the maximum value of the turn rate is reached for initial conditions giving Ntot = 70
efolds, along with a logarithmic fit. (4) The turn rate as a function of time for different
100
0.01
values of the initial angular velocity, with φ̇(0) = 0 and √
≤ γ̇(0) ≤ √
, in units of
2ξ
2ξ
√
τ = λ Mpl t.
198
In the long-wavelength limit, the adiabatic and entropic perturbations obey [38, 24]
k2
Ṙc = αHS + O
,
a2 H 2
2 k
,
Ṡ = βHS + O
2
a H2
(5.68)
so that we may define the transfer functions
Z
t
dt0 α(t0 )H(t0 )TSS (t∗ , t0 ),
t∗
Z t
0
0
0
TSS (t∗ , t) = exp
dt β(t )H(t ) .
TRS (t∗ , t) =
(5.69)
t∗
We take t∗ to be the time when a fiducial scale of interest first crossed the Hubble
radius during inflation, defined by a2 (t∗ )H 2 (t∗ ) = k∗2 . In [24], we calculated
α(t) =
2ω(t)
H(t)
(5.70)
and
β(t) = −2 − ηss + ησσ −
4 ω2
,
3 H2
(5.71)
where ≡ −Ḣ/H 2 and the other slow-roll parameters are defined as
Mσσ
,
V
ωI ω J MIJ
ηss ≡ Mpl2
.
ω2V
ησσ ≡ Mpl2
(5.72)
The dimensionless power spectrum is given by
k3
PR = 2 |Rc |2
2π
(5.73)
and hence, from Eqs. (5.68) and (5.69),
2
PR (k) = PR (k∗ ) 1 + TRS
(t∗ , t) ,
199
(5.74)
where k corresponds to a length scale that crossed the Hubble radius at some time
t > t∗ . The spectral index is then given by
ns (t) = ns (t∗ ) − [α(t) + β(t)TRS (t, t∗ )] sin(2∆),
(5.75)
TRS
cos ∆ ≡ p
.
2
1 + TRS
(5.76)
where
In the limit (ω/H) ησσ , the spectral index evaluated at N∗ assumes the single-field
form [29, 30, 34],
ns (t∗ ) = 1 − 6(t∗ ) + 2ησσ (t∗ ).
(5.77)
Crucial to note is that the turn rate, ω, serves as a window function within
TRS (t, t∗ ): once the coefficient α = 2ω/H becomes negligible, there will effectively be
no transfer of power from the entropic to the adiabatic perturbations, much as we
had found by examining the source terms on the righthand sides of Eqs. (5.20) and
(5.21). The question then becomes whether ω(t), and hence TRS (t∗ , t), can depart
appreciably from zero at times when perturbations on length scales of observational
interest first cross the Hubble radius.
The longest length scales of interest are often taken to be those that first crossed
the Hubble radius N∗ = 55 ± 5 efolds before the end of inflation [28, 29, 30]. Closer
analysis suggests that length scales that first crossed the Hubble radius N∗ = 62 − 63
efolds before the end of inflation correspond to the size of the present horizon [39].
Meanwhile, we follow [29] in assuming that successful inflation requires Ntot ≥ 70
efolds to solve the horizon and flatness problems. The question then becomes whether
ω(t), and hence TRS (t∗ , t), can differ appreciably from zero for N∗ ≤ 63. Given the
analysis in Section IV, the best chance for this to occur is for initial conditions that
produce the minimum amount of inflation, Ntot = 70.
In Table I, we present numerical results for key measures of multifield dynamics.
In each case we set ξ = 104 , φ(0) = 0.1Mpl , and χ(0) = 0. We vary χ̇(0) as shown
and adjust φ̇(0) in each case so as to produce exactly Ntot = 70 efolds of inflation.
200
χ̇(0)
10−2
10−1
1
101
102
ω(N∗ = 63)
1.16 × 10−10
1.20 × 10−9
9.41 × 10−9
1.18 × 10−7
1.12 × 10−6
TRS (max)
2.68 × 10−6
2.76 × 10−5
2.18 × 10−4
2.72 × 10−3
2.59 × 10−2
ns (N∗ = 63) ns (N∗ = 60)
0.969
0.967
0.969
0.967
0.969
0.967
0.969
0.967
0.973
0.967
Table 5.1: Numerical results for measures
√ of multifield dynamics for Higgs inflation with
ξ = 104 . We use dimensionless time τ =
λMpl t.
Because TRS remains so small in each of these cases, there is no discernible running
of the spectral index within the window N∗ = 63 to N∗ = 40 efolds before the end
of inflation. If we consider a fiducial scale k∗ that first crosses the Hubble radius
at N∗ = 63 efolds before the end of inflation, then we find ns = 0.97 across the
whole range of initial conditions, in excellent agreement with the measured value of
ns = 0.971 ± 0.010 [36]. If instead we set k∗ as the scale that first crossed the Hubble
radius N∗ = 60 efolds before the end of inflation, we find ns = 0.967 across the entire
range of initial conditions, again in excellent agreement with the latest measurements.
5.6
Conclusions
In this chapter we have analyzed Higgs inflation as a multifield model with nonminimal couplings. Because the Goldstone modes of the Standard Model electroweak
Higgs sector remain in the spectrum at high energies in renormalizable gauges, we
have incorporated their effects in the dynamics of the model. Because of the high
symmetry of the Higgs sector — guaranteed by the SU (2) electroweak gauge symmetry, which manifests as an SO(4) symmetry among the scalar fields of the Higgs
sector — the nonmiminal couplings for the various scalar fields take precisely the
same value (ξφ = ξχ = ξ), as do the tree-level couplings in the Jordan-frame potential
(λφ = λχ = λ, and so on). The effective potential in the Einstein frame therefore contains none of the irregular features, such as bumps or ridges, that were highlighted in
[24] for the case of multiple fields with arbitrary couplings. With no features such as
ridges off of which the fields may fall during their evolution, Hubble drag will always
201
cause any initial angular motion within field space to damp out rapidly. Increasing
the initial angular velocity to arbitrarily large values — well into what we call the
nonlinear regime — only increases the value of H at early times, which makes the
Hubble drag even more effective and hence hastens the damping out of the multifield
effects.
The rapidity with which the turn-rate damps to zero combined with the requirement of Ntot ≥ 70 efolds for successful inflation means that the multifield dynamics
become negligible before perturbations on scales of observational relevance first cross
the Hubble radius. Even if we push the observational window of interest back to
N∗ = 63 efolds before the end of inflation, rather than the usual assumption of
N∗ = 55 ± 5, we find that the model relaxes to effectively single-field dynamics prior
to N∗ . Hence the predictions from Higgs inflation for observable quantities, such as
the spectral index of the power spectrum of primordial perturbations, reduce to their
usual single-field form. Moreover, the absence of multifield effects for times later
than N∗ means that this model should produce negligible non-Gaussianities during
inflation, in contrast to the broader family of models studied in [24].
The methods we introduce here may be applied to any multifield model with
nonminimal couplings and an SO(N ) symmetry among the fields in field space. The
conclusion therefore appears robust that such highly symmetric models should behave
effectively as single-field models, at least within the observational window of interest
between N∗ = 63 and N∗ = 40 efolds before the end of inflation. Of course, multfield
effects could become important in such models at the end of inflation, during epochs
such as preheating [40]. Such processes remain under study.
5.7
Acknowledgements
It is a pleasure to thank Alan Guth, Mustafa Amin, and Leo Stein for helpful discussions. This work was supported in part by the U.S. Department of Energy (DoE)
under contract No. DE-FG02-05ER41360.
202
5.8
5.8.1
Appendix
Angular Evolution of the Field
For completeness, let us integrate the angular equation of motion, Eq. (5.53), for all
values of γ̇ (in the slow roll regime of the radial field). This yields
√
q
" √
#
γ̇(t)
λMpl + 2ξ γ̇02 + λMpl2
3λ Mpl t
= exp −
√
q
.
2ξ
γ̇0
λMpl + 2ξ γ̇ 2 (t) + λMpl2
In the two limits, γ̇0 √
(5.78)
√
√
√
λMpl / 2ξ and γ̇0 λMpl / 2ξ, we may solve Eq. (5.78)
and recover the forms of γ(t) presented in Eqs. (5.55) and (5.57).
203
204
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210
Chapter 6
Multifield Inflation after Planck:
The Case for Nonminimal
Couplings
Multifield models of inflation with nonminimal couplings are in excellent agreement
with the recent results from Planck. Across a broad range of couplings and initial
conditions, such models evolve along an effectively single-field attractor solution and
predict values of the primordial spectral index and its running, the tensor-to-scalar
ratio, and non-Gaussianities squarely in the observationally most-favored region. Such
models also can amplify isocurvature perturbations, which could account for the low
power recently observed in the CMB power spectrum at low multipoles. Future
measurements of primordial isocurvature perturbations could distinguish between the
currently viable possibilities.
6.1
Introduction
Early-universe inflation remains the leading framework for understanding a variety
of features of our observable universe [1, 2]. Most impressive has been the prediction
of primordial quantum fluctuations that could seed large-scale structure. Recent
measurements of the spectral tilt of primordial (scalar) perturbations, ns , find a
211
decisive departure from a scale-invariant spectrum [3, 4]. The Planck collaboration’s
value, ns = 0.9603 ± 0.0073, differs from ns = 1 by more than 5σ. At the same time,
observations with Planck constrain the ratio of tensor-to-scalar perturbations to r <
0.11 (95% CL), and are consistent with the absence of primordial non-Gaussianities,
fNL ∼ 0 [4, 5].
The Planck team also observes less power in the angular power spectrum of temperature anisotropies in the cosmic microwave background radiation (CMB) at low
multipoles, ` ∼ 20 − 40, compared to best-fit ΛCDM cosmology: a 2.5 − 3σ departure
on large angular scales, θ > 5◦ [6]. Many physical processes might ultimately account
for the deviation, but a primordial source seems likely given the long length-scales affected. One plausible possibility is that the discrepancy arises from the amplification
of isocurvature modes during inflation [4].
In this brief chapter we demonstrate that simple, well-motivated multifield models
with nonminimal couplings match the latest observations particularly well, with no
fine-tuning. This class of models (i) generically includes potentials that are concave
rather than convex at large field values, (ii) generically predicts values of r and ns in
the most-favored region of the recent observations. (iii) generically predicts fNL ∼ 0
except for exponentially fine-tuned initial field values, (iv) generically predicts ample
entropy production at the end of inflation, with an effective equation of state weff ∼
[0, 1/3], and (v) generically includes isocurvature perturbations as well as adiabatic
perturbations, which might account for the low power in the CMB power spectrum
at low multipoles.
We consider this class of models to be well-motivated for several reasons. Realistic
models of particle physics include multiple scalar fields at high energies. In any
such model, nonminimal couplings are required for self-consistency, since they arise
as renormalization counterterms when quantizing scalar fields in curved spacetime
[7]. Moreover, the nonminimal coupling constants generically rise with energy under
renormalization-group flow with no UV fixed-point [8], and hence one expects |ξ| 1
at inflationary energy scales. In such models inflation occurs for field values and
energy densities well below the Planck scale (see [9, 10, 11] and references therein).
212
Higgs inflation [11] is an elegant example: in renormalizable gauges (appropriate for
high energies) the Goldstone modes remain in the spectrum, yielding a multifield
model [10, 12, 13].
We demonstrate here for the first time that models of this broad class exhibit an
attractor behavior: over a wide range of couplings and fields’ initial conditions, the
fields evolve along an effectively single-field trajectory for most of inflation. Although
attractor behavior is common for single-field models of inflation [14], the dynamics
of multifield models generally show strong sensitivity to couplings and initial conditions (see, e.g., [15] and references therein). Not so for the class of multifield models
examined here, thanks to the shape of the effective potential in the Einstein frame.
The multifield attractor behavior demonstrated here means that for most regions of
phase space and parameter space, this general class of models yields values of ns ,
r, the running of the spectral index α = dns /d ln k, and fNL in excellent agreement
with recent observations. The well-known empirical success of single-field models
with nonminimal couplings [16, 11] is thus preserved for more realistic models involving multiple fields. Whereas the attractor behavior creates a large observational
degeneracy in the r vs. ns plane, the isocurvature spectra from these models depend
sensitively upon couplings and initial conditions. Future measurements of primordial
isocurvature spectra could therefore distinguish among models in this class.
6.2
Multifield Dynamics
In the Jordan frame, the fields’ nonminimal couplings remain explicit in the action,
Z
SJ =
p
1
I
µν
I
J
I
d x −g̃ f (φ )R̃ − δIJ g̃ ∂µ φ ∂ν φ − Ṽ (φ ) ,
2
4
(6.1)
where quantities in the Jordan frame are marked by a tilde. Performing the usual
conformal transformation, g̃µν (x) → gµν (x) = 2Mpl−2 f (φI (x)) g̃µν (x), where Mpl ≡
(8πG)−1/2 = 2.43 × 1018 GeV is the reduced Planck mass, we may write the action in
213
the Einstein frame as [9]
Z
SE =
Mpl2
1
µν
I
J
I
R − GIJ g ∂µ φ ∂ν φ − V (φ ) .
d x −g
2
2
4
√
(6.2)
The potential in the Einstein frame, V (φI ), is stretched by the conformal factor
compared to the Jordan-frame potential:
Mpl4
V (φ ) = 2 I Ṽ (φI ).
4f (φ )
I
(6.3)
The nonminimal couplings induce a curved field-space manifold in the Einstein frame
with metric GIJ (φK ) = (Mpl2 /(2f ))[δIJ +3f,I f,J /f ], where f,I = ∂f /∂φI [9]. We adopt
the form for f (φI ) required for renormalization [7],
#
"
X
1
ξI (φI )2 .
f (φI ) =
Mpl2 +
2
I
(6.4)
Here we consider two-field models, I, J = φ, χ.
As emphasized in [11, 10, 9], the conformal stretching of the Einstein-frame potential, Eq. (6.3), generically leads to concave potentials at large field values, even for
Jordan-frame potentials that are convex. In particular, for a Jordan-frame potential
of the simple form Ṽ (φI ) =
λφ 4
φ + g2 φ2 χ2 + λ4χ χ4 ,
4
Eqs. (6.3) and (6.4) yield a potential
in the Einstein frame that is nearly flat for large field values, V (φI ) → λJ Mpl4 /(4ξJ2 )
(no sum on J), as the Jth component of φI becomes arbitrarily large. This basic feature leads to “extra”-slow-roll evolution of the fields during inflation. If the couplings
λJ and ξJ are not equal to each other, V (φI ) develops ridges separated by valleys [9].
Inflation occurs in the valleys as well as along the ridges, since both are regions of
false vacuum with V 6= 0. See Fig. 1.
Constraints on r constrain the energy scale of inflation, H(t∗ )/Mpl < 3.7 × 10−5
[4]. For Higgs inflation, with λI = g = λφ and ξI = ξφ , the Hubble parameter during
q
slow roll is given by H/Mpl ' λφ /(12ξφ2 ). Measurements of the Higgs mass near
the electroweak symmetry-breaking scale require λφ ' 0.13. Under renormalizationgroup flow, λφ will fall to the range 0 < λφ < 0.01 at the inflationary energy scale;
214
Figure 6-1: Potential in the Einstein frame, V (φI ). Left: λχ = 0.75 λφ , g = λφ , ξχ = 1.2 ξφ .
Right: λχ = g = λφ , ξφ = ξχ . In both cases, ξI 1 and 0 < λI , g < 1.
λφ = 0.01 requires ξφ ≥ 780 to satisfy the constraint on H(t∗ )/Mpl , which in turn
requires ξφ ∼ O(101 − 102 ) at low energies [17]. For our general class of models, we
therefore consider couplings at the inflationary energy scale of order λI , g ∼ O(10−2 )
and ξI ∼ O(103 ) [18].
Expanding the scalar fields to first order, φI (xµ ) = ϕI (t) + δφI (xµ ), we find [9, 10]
σ̇ 2 = GIJ ϕ̇I ϕ̇J =
Mpl2
2f
"
#
˙2
3
f
φ̇2 + χ̇2 +
.
f
(6.5)
We also expand the spacetime metric to first order around a spatially flat FriedmannRobertson-Walker metric. Then the background dynamics are given by [9]
1
1 2
1 2
H =
σ̇ + V , Ḣ = −
σ̇ ,
2
3Mpl 2
2Mpl2
2
(6.6)
Dt ϕ̇I + 3H ϕ̇I + G IK V,K = 0,
where Dt is the (covariant) directional derivative, Dt AI ≡ ϕ̇J DJ AI = ȦI + ΓIJK AJ ϕ̇K
[19, 9]. The gauge-invariant Mukhanov-Sasaki variables for the linearized perturbations, QI , obey an equation of motion with a mass-squared matrix given by [19, 9]
MIJ ≡ G IJ (DJ DK V ) − RILM J ϕ̇L ϕ̇M ,
where RILM J is the Riemann tensor for the field-space manifold.
215
(6.7)
To analyze inflationary dynamics, we use a multifield formalism (see [2, 20] for
reviews) made covariant with respect to the nontrivial field-space curvature (see [9, 19]
and references therein). We define adiabatic and isocurvature directions in the curved
field space via the unit vectors σ̂ I ≡ ϕ̇I /σ̇ and ŝI ≡ ω I /ω, where the turn-rate vector
is given by ω I ≡ Dt σ̂ I , and ω = |ω I |. We also define slow-roll parameters [9, 19]:
≡−
J
I
J
I
Ḣ
2 ŝI ŝ M J
2 σ̂I σ̂ M J
,
η
≡
M
.
,
η
≡
M
ss
σσ
pl
pl
H2
V
V
(6.8)
Using Eq. (6.6), we have the exact relation, = 3σ̇ 2 /(σ̇ 2 + 2V ). The adiabatic
and isocurvature perturbations may be parameterized as Rc = (H/σ̇)σ̂I QI and S =
(H/σ̇)ŝI QI , where Rc is the gauge-invariant curvature perturbation. Perturbations of
pivot-scale k∗ = 0.002 Mpc−1 first crossed outside the Hubble radius during inflation
at time t∗ . In the long-wavelength limit, the evolution of Rc and S for t > t∗ is given
by the transfer functions [9, 19]
Z
t
dt0 2ω(t0 )TSS (t∗ , t0 ),
t∗
Z t
0
0
0
dt β(t )H(t ) ,
TSS (t∗ , t) = exp
TRS (t∗ , t) =
(6.9)
t∗
2
with β(t) = −2 − ηss + ησσ − 34 Hω 2 . Given the form of TRS , the perturbations Rc and
S decouple if ω I = 0.
The dimensionless power spectrum for the adiabatic perturbations is defined as
PR (k) = (2π)−2 k 3 |Rc |2 and the spectral index is defined as ns − 1 ≡ ∂ ln PR /∂ ln k.
Around t∗ the spectral index is given by [9, 2, 20, 19]
ns (t∗ ) = 1 − 6(t∗ ) + 2ησσ (t∗ ).
(6.10)
At late times and in the long-wavelength limit, the power spectrum becomes PR =
2
PR (k∗ ) [1 + TRS
], and hence the spectral index may be affected by the transfer of
power from isocurvature to adiabatic modes: ns (t) = ns (t∗ )+H∗−1 (∂TRS /∂t∗ ) sin(2∆),
2
with cos ∆ ≡ TRS (1 + TRS
)−1/2 .
216
The mass of the isocurvature perturbations is µ2s = 3H 2 (ηss + ω 2 /H 2 ) [9]. For
2
µs < 3H/2 we have PS (k∗ ) ' PR (k∗ ) and hence PS ' PR (t∗ )TSS
at late times. In
the Einstein frame the anisotropic pressure Πi j ∝ T ij for i 6= j vanishes to linear
order, so the tensor perturbations hij evolve just as in single-field models with PT '
2
128 [H(t∗ )/Mpl ]2 (k/k∗ )−2 , and therefore r ≡ PT /PR = 16/[1 + TRS
] [2, 20, 19].
6.3
The Single-Field Attractor
To study the single-field attractor behavior, we first consider the case in which the
system inflates in a valley along the χ = 0 direction, perhaps after first rolling off a
ridge. In the slow-roll limit and with χ ∼ χ̇ ∼ 0, Eq. (6.6) reduces to [10]
φ̇SR
Using H/Mpl '
p
λφ Mpl3
'− √ 2 .
3 3 ξφ φ
(6.11)
q
λφ /(12ξφ2 ) we may integrate Eq. (6.11),
4
ξφ φ2∗
' N∗ ,
2
Mpl
3
(6.12)
where N∗ is the number of efolds before the end of inflation, and we have used φ(t∗ ) φ(tend ). (We arrive at comparable expressions if the system falls into a valley along
some angle in field space, θ ≡ arctan (φ/χ).) Eq. (6.5) becomes σ̇ 2 |χ=0 ' 6Mpl2 φ̇2 /φ2
upon using ξφ 1. Using V ' λφ Mpl4 /(4ξφ2 ) and Eqs. (6.11), (6.12) in Eq. (6.8) we
find
'
3
.
4N∗2
(6.13)
To estimate ησσ we use ησσ = − σ̈/(H σ̇) + O(2 ) [2], and find
ησσ
1
'−
N∗
3
1−
4N∗
.
(6.14)
All dependence on λI and ξI has dropped out of these expressions for and ησσ
in Eqs. (6.13) and (6.14). For a broad range of initial field values and velocities —
217
and independent of the couplings — this entire class of models should quickly relax
into an attractor solution in which the fields evolve along an effectively single-field
trajectory with vanishing turn-rate, ω I ∼ 0. Within this attractor solution we find
analytically ∗ = 2.08 × 10−4 and ησσ∗ = −0.0165 for N∗ = 60; and ∗ = 3.00 × 10−4
and ησσ∗ = −0.0197 for N∗ = 50. To test this attractor behavior, we performed
numerical simulations with a sampling of couplings and initial conditions. We fixed
λφ = 0.01 and ξφ = 103 and looped over λχ = {0.5, 0.75, 1} λφ , g = {0.5, 0.75, 1} λφ ,
and ξχ = {0.8, 1, 1.2} ξφ . These parameters gave a variety of potentials with combinations of ridges and valleys along different directions in field space. We set the initial
p
−1/2 −1/2
amplitude of the fields to be φ20 + χ20 = 10×max[ξφ , ξχ ] (in units of Mpl ), which
generically produced 70 or more efolds of inflation. We varied the initial angle in field
space, θ0 = arctan (φ0 /χ0 ), among the values θ0 = {0, π/6, π/3, π/2}, and allowed for
a relatively wide range of initial fields velocities: φ̇0 , χ̇0 = {−10 |φ̇SR |, 0, +10 |φ̇SR |},
where φ̇SR is given by Eq. (6.11).
Typical trajectories are shown in Fig. 6-2a. In each case, the fields quickly rolled
into a valley and, after a brief, transient period of oscillation, evolved along a straight
trajectory in field-space for the remainder of inflation with ω I = 0. Across this entire
range of couplings and initial conditions, the analytic expressions for and ησσ in
Eqs. (6.13)-(6.14) provide close agreement with the exact numerical simulations. See
Fig. 2b.
We confirmed numerically that for much larger initial field velocities, up to φ̇0 , χ̇0 ∼
106 |φ̇SR |, such that the initial kinetic energy is larger than the difference between
ridge-height and valley in the potential, the system exhibits a very brief, transient
period of rapid angular motion (akin to [10]). The fields’ kinetic energy rapidly redshifts away so that the fields land in a valley of the potential within a few efolds,
after which slow-roll inflation continues along a single-field attractor trajectory just
like the ones shown in Fig. 6-2a. Moreover, the attractor behavior is unchanged
if one considers bare masses mφ , mχ Mpl or a negative coupling g < 0, so long
p
as one imposes the fairly minimal constraint that V ≥ 0 and hence g > − λφ λχ .
(Each of these features could affect preheating dynamics but not the attractor be218
Figure 6-2: Left: Field trajectories for different couplings and initial conditions (here for
φ̇0 , χ̇0 = 0). Open circles indicate fields’ initial values. The parameters {λχ , g, ξχ , θ0 } are
given by: {0.75λφ , λφ , 1.2ξφ , π/4} (red), {λφ , λφ , 0.8ξφ , π/4} (blue), {λφ , 0.75λφ , 0.8ξφ , π/6}
(green), {λφ , 0.75 λφ , 0.8ξφ , π/3} (black). Right: Numerical vs. analytic evaluation of the
slow-roll parameters, (numerical = blue, analytic = red) and ησσ (numerical = black,
analytic = green), for λφ = 0.01, λχ = 0.75 λφ , g = λφ , ξφ = 103 , and ξχ = 1.2 ξφ , with
θ0 = π/4 and φ̇0 = χ̇0 = +10 |φ̇SR |.
havior during inflation.) Lastly, we performed numerical simulations for the case of
three fields rather than two, and again found that the dynamics quickly relax to the
single-field attractor since the effective potential contains ridges and valleys, so the
fields generically wind up within a valley.
6.4
Observables and the Attractor Solution
As we have confirmed numerically, trajectories in the single-field attractor solution
generically have ω I ∼ 0 between t∗ and tend (which we define as (tend ) = 1, or
ä(tend ) = 0); hence TRS ∼ 0 for these trajectories. The spectral index ns (t) therefore
2
reduces to ns (t∗ ) of Eq. (6.10), and r reduces to r = 16 [1 + O(TRS
)] ' 16. Using
Eqs. (6.13) and (6.14), we then find
ns ' 1 −
2
3
12
− 2, r ' 2,
N∗ N∗
N∗
(6.15)
and hence ns = 0.966 and r = 0.0033 for N∗ = 60; and ns = 0.959 and r = 0.0048
for N∗ = 50. We also calculated ns and r numerically for each of the trajectories
219
described above, and found ns = 0.967 and r = 0.0031 for N∗ = 60, and ns = 0.960
and r = 0.0044 for N∗ = 50. These values sit right in the most-favored region of the
latest observations. (See Fig. 1 in [4].) Even for a low reheat temperature, we find
ns within 2σ of the Planck value for N∗ ≥ 38. The predicted value r ∼ 10−3 could be
tested by upcoming CMB polarization experiments.
For the running of the spectral index, α ≡ dns /d ln k, we use Eq. (6.15), the
Rt
general relationship (dx/d ln k) |∗ ' (ẋ/H) |∗ [2], and N∗ = Ntot − ti∗ Hdt to find
dns
2
α=
'− 2
d ln k
N∗
3
1+
,
N∗
(6.16)
which yields α = −5.83 × 10−4 for N∗ = 60 and α = −8.48 × 10−4 for N∗ = 50,
fully consistent with the result from Planck, α = −0.0134 ± 0.0090, indicating no
observable running of the spectral index [4].
Meanwhile, for every trajectory in our large sample we numerically computed fNL
following the methods of [9]. Across the whole range of couplings and initial conditions
considered here, we found |fNL | < 0.1, consistent with the latest observations [5]. In
these models fNL is exponentially sensitive to the fields’ initial conditions, requiring
a fine-tuning of O(10−4 ) to produce |fNL | > 1 [9]. In the absence of such fine-tuning
these models generically predict |fNL | O(1).
Unlike several models with concave potentials analyzed in [4], multifield models
with nonminimal couplings should produce entropy efficiently at the end of inflation,
when ξI (φI )2 < Mpl2 . The energy density and pressure are given by ρ = 21 σ̇ 2 + V (φI )
and p =
1 2
σ̇
2
− V (φI ) [9]. We confirmed numerically that for every trajectory in
our large sample, the effective equation of state w = p/ρ averaged to 0 beginning at
tend (when = 1) and asymptoted to 1/3 within a few oscillations. This behavior
may be understood analytically from the virial theorem, which acquires corrections
proportional to gradients of the field-space metric coefficients, just like applications
in curved spacetime [21]. We find hσ̇ 2 i = hV,J ϕJ i + hCi = h2Mpl4 V /f i + hCi, where
C ≡ − 12 (∂J GKL )ϕ̇K ϕ̇L ϕJ . More generally, inflation in these models ends with one or
both fields oscillating quasi-periodically around the minimum of the potential, and
220
hence preheating should be efficient [2, 22, 23].
6.5
Isocurvature Perturbations
The models in this class predict three basic possibilities for isocurvature perturbations,
depending on whether inflation occurs while the fields are in a valley, on top of a ridge,
or in a symmetric potential with λI = g = λ and ξI = ξ and hence no ridges (like
2
2
2
Higgs inflation). The fraction βiso (k) ≡ PS (k)/[PR (k) + PS (k)] = TSS
/[1 + TRS
+ TSS
]
[4] may distinguish between the various situations. In each of these scenarios, ω I ∼
0 and hence TRS ∼ 0. Inflating in a valley, ηss > 1 so µ2s /H 2 > 9/4 and the
(heavy) isocurvature modes are suppressed, TSS → 0 and hence βiso ∼ 0 for scales k
corresponding to N∗ = 60 − 50. Inflating on top of a ridge, ηss < 0 so µ2s /H 2 < 0 and
the isocurvature modes grow via tachyonic instability, TSS 1, and hence βiso ∼ 1
across the same scales k. Scenarios in which the fields begin on top of a ridge and roll
off at intermediate times can give any value 0 ≤ βiso ≤ 1 depending sensitively upon
initial conditions [24]. In the case of symmetric couplings, µ2s /H 2 ' 0 [10], yielding
TSS ∼ O(10−3 ) and βiso = 2.23 × 10−5 for N∗ = 60 and βiso = 3.20 × 10−5 for N∗ = 50
[25].
6.6
Conclusions
Multifield models of inflation with nonminimal couplings possess a strong singlefield attractor solution, such that they share common predictions for ns , r, α, fNL ,
and for efficient entropy production across a broad range of couplings and initial
conditions. The predicted spectral observables provide excellent agreement with the
latest observations. These models differ, however, in their predicted isocurvature
perturbation spectra, which might help break the observational degeneracy among
members of this class.
Note. While the paper, which the present chapter is based on, was under review,
similar results regarding attractor behavior in models with nonminimal couplings
221
were presented in [26].
6.7
Acknowledgements
It is a pleasure to thank Bruce Bassett, Rhys Borchert, Xingang Chen, Joanne Cohn,
Alan Guth, Carter Huffman, Edward Mazenc, and Katelin Schutz for helpful discussions. This work was supported in part by the U.S. Department of Energy (DoE)
under contract No. DE-FG02-05ER41360.
222
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227
228
Chapter 7
Multifield Inflation after Planck :
Isocurvature Modes from
Nonminimal Couplings
Recent measurements by the Planck experiment of the power spectrum of temperature
anisotropies in the cosmic microwave background radiation (CMB) reveal a deficit of
power in low multipoles compared to the predictions from best-fit ΛCDM cosmology.
If the low-` anomaly persists after additional observations and analysis, it might
be explained by the presence of primordial isocurvature perturbations in addition
to the usual adiabatic spectrum, and hence may provide the first robust evidence
that early-universe inflation involved more than one scalar field. In this chapter we
explore the production of isocurvature perturbations in nonminimally coupled twofield inflation. We find that this class of models readily produces enough power in
the isocurvature modes to account for the Planck low-` anomaly, while also providing
excellent agreement with the other Planck results.
7.1
Introduction
Inflation is a leading cosmological paradigm for the early universe, consistent with
the myriad of observable quantities that have been measured in the era of precision
229
cosmology [1, 2, 3]. However, a persistent challenge has been to reconcile successful
inflationary scenarios with well-motivated models of high-energy physics. Realistic
models of high-energy physics, such as those inspired by supersymmetry or string
theory, routinely include multiple scalar fields at high energies [4]. Generically, each
scalar field should include a nonminimal coupling to the spacetime Ricci curvature
scalar, since nonminimal couplings arise as renormalization counterterms when quantizing scalar fields in curved spacetime [5, 6, 7, 8]. The nonminimal couplings typically
increase with energy-scale under renormalization-group flow [7], and hence should be
large at the energy-scales of interest for inflation. We therefore study a class of inflationary models that includes multiple scalar fields with large nonminimal couplings.
It is well known that the predicted perturbation spectra from single-field models
with nonminimal couplings produce a close fit to observations. Following conformal
transformation to the Einstein frame, in which the gravitational portion of the action
assumes canonical Einstein-Hilbert form, the effective potential for the scalar field is
stretched by the conformal factor to be concave rather than convex [9, 10], precisely
the form of inflationary potential most favored by the latest results from the Planck
experiment [11].
The most pronounced difference between multifield inflation and single-field inflation is the presence of more than one type of primordial quantum fluctuation that can
evolve and grow. The added degrees of freedom may lead to observable departures
from the predictions of single-field models, including the production and amplification
of isocurvature modes during inflation [12, 13, 14, 15, 16, 17, 18, 19].
Unlike adiabatic perturbations, which are fluctuations in the energy density, isocurvature perturbations arise from spatially varying fluctuations in the local equation
of state, or from relative velocities between various species of matter. When isocurvature modes are produced primordially and stretched beyond the Hubble radius,
causality prevents the redistribution of energy density on super-horizon scales. When
the perturbations later cross back within the Hubble radius, isocurvature modes create
pressure gradients that can push energy density around, sourcing curvature perturbations that contribute to large-scale anisotropies in the cosmic microwave background
230
radiation (CMB). (See, e.g., [20, 11].)
The recent measurements of CMB anisotropies by Planck favor a combination of
adiabatic and isocurvature perturbations in order to improve the fit at low multipoles
(` ∼ 20 − 40) compared to the predictions from the simple, best-fit ΛCDM model in
which primordial perturbations are exclusively adiabatic. The best fit to the present
data arises from models with a modest contribution from isocurvature modes, whose
primordial power spectrum PS (k) is either scale-invariant or slightly blue-tilted, while
the dominant adiabatic contribution, PR (k), is slightly red-tilted [11]. The low-`
anomaly thus might provide the first robust empirical evidence that early-universe
inflation involved more than one scalar field.
Well-known multifield models that produce isocurvature perturbations, such as
axion and curvaton models, are constrained by the Planck results and do not improve
the fit compared to the purely adiabatic ΛCDM model [11]. As we demonstrate here,
on the other hand, the general class of multifield models with nonminimal couplings
can readily produce isocurvature perturbations of the sort that could account for the
low-` anomaly in the Planck data, while also producing excellent agreement with the
other spectral observables measured or constrained by the Planck results, such as the
spectral index ns , the tensor-to-scalar ratio r, the running of the spectral index α,
and the amplitude of primordial non-Gaussianity fNL .
Nonminimal couplings in multifield models induce a curved field-space manifold
in the Einstein frame [21], and hence one must employ a covariant formalism for
this class of models. Here we make use of the covariant formalism developed in
[22], which builds on pioneering work in [13, 18]. In Section 7.2 we review the most
relevant features of our class of models, including the formal machinery required to
study the evolution of primordial isocurvature perturbations. In Section 7.3 we focus
on a regime of parameter space that is promising in the light of the Planck data,
and for which analytic approxmations are both tractable and in close agreement with
numerical simulations. In Section 7.4 we compare the predictions from this class of
models to the recent Planck findings. Concluding remarks follow in Section 7.5.
231
7.2
Model
We consider two nonminimally coupled scalar fields φI {φ, χ}. We work in 3+1
spacetime dimensions with the spacetime metric signature (−, +, +, +). We express
our results in terms of the reduced Planck mass, Mpl ≡ (8πG)−1/2 = 2.43 × 1018 GeV.
Greek letters (µ, ν) denote spacetime 4-vector indices, lower-case Roman letters (i,
j) denote spacetime 3-vector indices, and capital Roman letters (I, J) denote fieldspace indices. We indicate Jordan-frame quantities with a tilde, while Einstein-frame
quantities will be sans tilde. Subscripted commas indicate ordinary partial derivatives
and subscripted semicolons denote covariant derivatives with respect to the spacetime
coordinates.
We begin with the action in the Jordan frame, in which the fields’ nonminimal
couplings remain explicit:
Z
S̃ =
p
1
I
µν
I
J
I
d x −g̃ f (φ )R̃ − G̃IJ g̃ ∂µ φ ∂ν φ − Ṽ (φ ) ,
2
4
(7.1)
where R̃ is the spacetime Ricci scalar, f (φI ) is the nonminimal coupling function, and
G̃IJ is the Jordan-frame field space metric. We set G̃IJ = δIJ , which gives canonical
kinetic terms in the Jordan frame. We take the Jordan-frame potential, Ṽ (φI ), to
have a generic, renormalizable polynomial form with an interaction term:
Ṽ (φ, χ) =
λφ 4 g 2 2 λχ 4
φ + φχ + χ,
4
2
4
(7.2)
with dimensionless coupling constants λI and g. As discussed in [22], the inflationary
dynamics in this class of models are relatively insensitive to the presence of mass
terms, m2φ φ2 or m2χ χ2 , for realistic values of the masses that satisfy mφ , mχ Mpl .
Hence we will neglect such terms here.
232
7.2.1
Einstein-Frame Potential
We perform a conformal transformation to the Einstein frame by rescaling the spacetime metric tensor,
g̃µν (x) = Ω2 (x) gµν (x),
(7.3)
where the conformal factor Ω2 (x) is related to the nonminimal coupling function via
the relation
Ω2 (x) =
2
f φI (x) .
2
Mpl
(7.4)
This transformation yields the action in the Einstein frame,
Z
S=
Mpl2
1
µν
I
J
I
d x −g
R − GIJ g ∂µ φ ∂ν φ − V (φ ) ,
2
2
4
√
(7.5)
where all the terms sans tilde are stretched by the conformal factor. For instance,
the conformal transformation to the Einstein frame induces a nontrivial field-space
metric [21]
GIJ
Mpl2
3
=
δIJ + f,I f,J ,
2f
f
(7.6)
and the potential is also stretched so that it becomes
Mpl4
Ṽ (φ, χ)
(2f )2
Mpl4 λφ 4 g 2 2 λχ 4
=
φ + φχ + χ .
(2f )2 4
2
4
V (φ, χ) =
(7.7)
The form of the nonminimal coupling function is set by the requirements of renormalization [5, 6],
1
f (φ, χ) = [M 2 + ξφ φ2 + ξχ χ2 ],
2
(7.8)
where ξφ and ξχ are dimensionless couplings and M is some mass scale such that when
the fields settle into their vacuum expectation values, f → Mpl2 /2. Here we assume
that any nonzero vacuum expectation values for φ and χ are much smaller than the
Planck scale, and hence we may take M = Mpl .
The conformal stretching of the potential in the Einstein frame makes it concave
233
Figure 7-1: Potential in the Einstein frame, V (φI ) in Eq. (7.7). The parameters
shown here are λχ = 0.75 λφ , g = λφ , ξχ = 1.2 ξφ , with ξφ 1 and λφ > 0.
and asymptotically flat along either direction in field space, I = φ, χ,
Mpl4 λI
Mpl2
V (φ ) →
1+O
4 ξI2
ξJ (φI )2
I
(7.9)
(no sum on I). For non-symmetric couplings, in which λφ 6= λχ and/or ξφ 6= ξχ , the
potential in the Einstein frame will develop ridges and valleys, as shown in Fig. 7-1.
p
Crucially, V > 0 even in the valleys (for g > − λφ λχ ), and hence the system will
inflate (albeit at varying rates) whether the fields ride along a ridge or roll within a
valley, until the fields reach the global minimum of the potential at φ = χ = 0.
Across a wide range of couplings and initial conditions, the models in this class
obey a single-field attractor [19]. If the fields happen to begin evolving along the top
of a ridge, they will eventually fall into a neighboring valley. Motion in field space
transverse to the valley will quickly damp away (thanks to Hubble drag), and the
fields’ evolution will include almost no further turning in field space. Within that
single-field attractor, predictions for ns , r, α, and fNL all fall squarely within the
most-favored regions of the latest Planck measurements [19].
The fields’ approach to the attractor behavior — essentially, how quickly the fields
roll off a ridge and into a valley — depends on the local curvature of the potential
near the top of a ridge. Consider, for example, the case in which the direction χ = 0
corresponds to a ridge. To first order, the curvature of the potential in the vicinity
234
ns
1.0
0.5
0.0
0.01
0.1
1
Κ
10
100
Figure 7-2: The spectral index ns (red), as given in Eq. (7.61), for different values
of κ, which characterizes the local curvature of the potential near the top of a ridge.
Also shown are the 1σ (thin, light blue) and 2σ (thick, dark blue) bounds on ns from
the Planck measurements. The couplings shown here correspond to ξφ = ξχ = 103 ,
λφ = 10−2 , and λχ = g, fixed for a given value of κ from Eq. (7.10). The fields’ initial
conditions are φ = 0.3, φ̇0 = 0, χ0 = 10−3 , χ̇0 = 0, in units of Mpl .
of χ = 0 is proportional to (gξφ − λφ ξχ ) [22]. As we develop in detail below, a
convenient combination with which to characterize the local curvature near the top
of such a ridge is
κ≡
4(λφ ξχ − gξφ )
.
λφ
(7.10)
As shown in Fig. 7-2, models in this class produce excellent agreement with the
latest measurements of ns from Planck across a wide range of parameters, where ns ≡
1 + d ln PR /d ln k. Strong curvature near the top of the ridge corresponds to κ 1:
in that regime, the fields quickly roll off the ridge, settle into a valley of the potential,
and evolve along the single-field attractor for the duration of inflation, as analyzed in
[19]. More complicated field dynamics occur for intermediate values, 0.1 < κ < 4, for
which multifield dynamics pull ns far out of agreement with empirical observations.
The models again produce excellent agreement with the Planck measurements of ns
in the regime of weak curvature, 0 ≤ κ ≤ 0.1.
As we develop below, other observables of interest, such as r, α, and fNL , likewise show excellent fit with the latest observations. In addition, the regime of weak
235
curvature, κ 1, is particularly promising for producing primordial isocurvature
perturbations with characteristics that could explain the low-` anomaly in the recent
Planck measurements. Hence for the remainder of this chapter we focus on the regime
κ 1, a region that is amenable to analytic as well as numerical analysis.
7.2.2
Coupling Constants
The dynamics of this class of models depend upon combinations of dimensionless
coupling constants like κ defined in Eq. (7.10) and others that we introduce below.
The phenomena analyzed here would therefore hold for various values of λI and ξI ,
such that combinations like κ were unchanged. Nonetheless, it is helpful to consider
reasonable ranges for the couplings on their own.
The present upper bound on the tensor-to-scalar ratio, r < 0.12, constrains the
energy-scale during inflation to satisfy H(thc )/Mpl ≤ 3.7 × 10−5 [11], where H(thc )
is the Hubble parameter at the time during inflation when observationally relevant
perturbations first crossed outside the Hubble radius. During inflation the dominant
contribution to H will come from the value of the potential along the direction in
which the fields slowly evolve. Thus we may use the results from Planck and Eq.
(7.9) to set a basic scale for the ratios of couplings, λI /ξI2 . For example, if the fields
evolve predominantly along the direction χ ∼ 0, then during slow roll the Hubble
parameter will be
s
H'
λφ
Mpl ,
12ξφ2
(7.11)
and hence the constraint from Planck requires λφ /ξφ2 ≤ 1.6 × 10−8 .
We adopt a scale for the self-couplings λI by considering a particularly elegant
member of this class of models. In Higgs inflation [10], the self-coupling λφ is fixed
by measurements of the Higgs mass near the electroweak symmetry-breaking scale,
λφ ' 0.1, corresponding to mH ' 125 GeV [23, 24]. Under renormalization-group
flow, λφ will fall to the range 0 < λφ < 0.01 at the inflationary energy scale [28]. Eq.
(7.11) with λ = 0.01 requires ξφ ≥ 780 at inflationary energy scales to give the correct
amplitude of density perturbations. For our general class of models, we therefore
236
consider couplings at the inflationary energy scale of order λI , g ∼ O(10−2 ) and ξI ∼
O(103 ). Taking into account the running of both λI and ξI under renormalizationgroup flow, these values correspond to λI ∼ O(10−1 ) and ξI ∼ O(102 ) at low energies
[28].
We consider these to be reasonable ranges for the couplings. Though one might
prefer dimensionless coupling constants to be O(1) in any “natural” scenario, the
ranges chosen here correspond to low-energy couplings that are no more fine-tuned
than the fine-structure constant, αEM ' 1/137. Indeed, our choices are relatively
conservative. For the case of Higgs inflation, the running of λφ is particularly sensitive
to the mass of the top quark. Assuming a value for mtop at the low end of the present
2σ bound yields λφ ' 10−4 rather than 10−2 at high energies, which in turn requires
ξφ ≥ 80 at the inflationary energy scale rather than ξφ ≥ 780 [29]. Nonetheless,
for illustrative purposes, we use λI , g ∼ 10−2 and ξI ∼ 103 for the remainder of our
analysis.
We further note that despite such large nonminimal couplings, ξI ∼ 103 , our
analysis is unhindered by any potential breakdown of unitarity. The energy scale
at which unitarity might be violated for Higgs inflation has occasioned a great deal
of heated debate in the literature, with conflicting claims that the renormalization
p
cut-off scale should be in the vicinity of Mpl , Mpl / ξφ , or Mpl /ξφ [30]. Even if one
adopted the most stringent of these suggested cut-off scales, Mpl /ξφ ∼ 10−3 Mpl ,
the relevant dynamics for our analysis would still occur at energy scales well below
the cut-off, given the constraint H(thc ) ≤ 3.7 × 10−5 Mpl . (The unitarity cut-off
scale in multifield models in which the nonminimal couplings ξI are not all equal to
each other has been considered in [31], which likewise identify regimes of parameter
space in which Λeff remains well above the energy scales and field values relevant to
inflation.) Moreover, models like Higgs inflation can easily be “unitarized” with the
addition of a single heavy scalar field [32], and hence all of the following analysis could
be considered the low-energy dynamics of a self-consistent effective field theory. The
methods developed here may be applied to a wide class of models, including those
studied in [39, 40, 41, 42].
237
Finally, we note that for couplings λI , g ∼ 10−2 and ξI ∼ 103 at high energies, the
regime of weak curvature for the potential, κ < 0.1, requires that the couplings be
close but not identical to each other. In particular, κ ∼ 0.1 requires g/λφ ∼ ξχ /ξφ ∼
1 ± O(10−5 ). Such small differences are exactly what one would expect if the effective
couplings at high energies arose from some softly broken symmetry. For example, the
field χ could couple to some scalar cold dark matter (CDM) candidate (perhaps a
supersymmetric partner) or to a neutrino, precisely the kinds of couplings that would
be required if the primordial isocurvature perturbations were to survive to late times
and get imprinted in the CMB [20]. In that case, corrections to the β functions for
the renormalization-group flow of the couplings λχ and ξχ would appear of the form
2
gX
/16π 2 [7, 33], where gX is the coupling of χ to the new field. For reasonable values
of gX ∼ 10−1 − 10−2 , such additional terms could easily account for the small but
non-zero differences among couplings at the inflationary energy scale.
7.2.3
Dynamics and Transfer Functions
When we vary the Einstein-frame action with respect to the fields φI , we get the
equations of motion, which may be written
φI + ΓIJK ∂µ φJ ∂ µ φK − G IJ V,K = 0,
(7.12)
where φI ≡ g µν φI;µ;ν and ΓIJK is the field-space Christoffel symbol.
We further expand each scalar field to first order in perturbations about its classical background value,
φI (xµ ) = ϕI (t) + δφI (xµ )
(7.13)
and we consider scalar perturbations to the spacetime metric (which we assume to
be a spatially flat Friedmann-Robertson-Walker metric) to first order:
ds2 = −(1 + 2A)dt2 + 2a(t)(∂i B)dxi dt+
a(t)2 [(1 − 2ψ)δij + 2∂i ∂j E]dxi dxj ,
238
(7.14)
where a(t) is the scale factor and A, B, ψ and E are the scalar gauge degrees of
freedom.
Under this expansion, the full equations of motion separate into background and
first-order equations. The background equations are given by
Dt ϕ̇I + 3H ϕ̇I + G IJ V,J = 0,
(7.15)
where DJ AI ≡ ∂J AI +ΓIJK AK for an arbitrary vector, AI , on the field-space manifold;
Dt AI ≡ ϕ̇J DJ AI is a directional derivative; and H ≡ ȧ/a is the Hubble parameter.
The 00 and 0i components of the background-order Einstein equations yield:
1
1
I J
I
GIJ ϕ̇ ϕ̇ + V (ϕ )
H =
3Mpl2 2
1
Ḣ = −
GIJ ϕ̇I ϕ̇J .
2Mpl2
2
(7.16)
Using the covariant formalism of [22], we find the equations of motion for the perturbations,
Dt2 QI + 3HDt QI +
"
3
#
1
a
k2 I
δ + MIJ − 2 3 Dt
ϕ̇I ϕ̇J
QJ = 0,
a2 J
Mpl a
H
(7.17)
where QI is the gauge-invariant Mukhanov-Sasaki variable
QI = QI +
ϕ̇I
ψ,
H
(7.18)
and QI is a covariant fluctuation vector that reduces to δφI to first order in the
fluctuations. Additionally, MIJ is the effective mass-squared matrix given by
MIJ ≡ G IK DJ DK V − RILM J ϕ̇L ϕ̇M ,
(7.19)
where RILM J is the field-space Riemann tensor.
The degrees of freedom of the system may be decomposed into adiabatic and
239
entropic (or isocurvature) by introducing the magnitude of the background fields’
velocity vector,
σ̇ ≡ |ϕ̇I | =
p
GIJ ϕ̇I ϕ̇J ,
(7.20)
with which we may define the unit vector
ϕ̇I
σ̇
σ̂ I ≡
(7.21)
which points along the fields’ motion. Another important dynamical quantity is the
turn-rate of the background fields, given by
ω I = Dt σ̂ I ,
(7.22)
with which we may construct another important unit vector,
ŝI ≡
ωI
,
ω
(7.23)
where ω = |ω I |. The vector ŝI points perpendicular to the fields’ motion, ŝI σ̂I = 0.
The unit vectors σ̂ I and ŝI effectively act like projection vectors, with which we may
decompose any vector into adiabatic and entropic components. In particular, we may
decompose the vector of fluctuations QI ,
Qσ ≡ σ̂I QI
(7.24)
I
Qs ≡ ŝI Q ,
in terms of which Eq. (7.17) separates into two equations of motion:
"
k2
1 d
Q̈σ + 3H Q̇σ + 2 + Mσσ − ω 2 − 2 3
a
Mpl a dt
!
d
V,σ Ḣ
= 2 (ω Qs ) − 2
+
ω Qs ,
dt
σ̇
H
a3 σ̇ 2
H
#
Qσ
k2
ω k2
2
Ψ,
Q̈s + 3H Q̇s + 2 + Mss + 3ω Qs = 4Mpl2
a
σ̇ a2
(7.25)
240
(7.26)
where Ψ is the gauge-invariant Bardeen potential [3],
B
,
Ψ ≡ ψ + a H Ė −
a
2
(7.27)
and where Mσσ and Mss are the adiabatic and entropic projections of the masssquared matrix, MIJ from (7.19). More explicitly,
Mσσ = σ̂I σ̂ J MIJ
(7.28)
Mss = ŝI ŝJ MIJ .
As Eqs. (7.25) and (7.26) make clear, the entropy perturbations will source the
adiabatic perturbations but not the other way around, contingent on the turn-rate ω
being nonzero. We also note that the entropy perturbations have an effective masssquared of
µ2s = Mss + 3ω 2 .
(7.29)
In the usual fashion [3], we may construct the gauge-invariant curvature perturbation,
Rc ≡ ψ −
H
δq
(ρ + p)
(7.30)
where ρ and p are the background-order density and pressure and δq is the energydensity flux of the perturbed fluid. In terms of our projected perturbations, we find
[22]
Rc =
H
Qσ .
σ̇
(7.31)
Analogously, we may define a normalized entropy (or isocurvature) perturbation as
[3, 13, 14, 16, 18, 22]
S≡
H
Qs .
σ̇
(7.32)
In the long-wavelength limit, the coupled perturbations obey relations of the form
[3, 13, 14, 16, 18, 22]:
Ṙc ' αHS
Ṡ ' βHS,
241
(7.33)
which allows us to write the transfer functions as
Z
t
dt0 α(t0 ) H(t0 ) TSS (thc , t0 )
thc
Z t
0
0
0
TSS (thc , t) = exp
dt β(t ) H(t ) ,
TRS (thc , t) =
(7.34)
thc
where thc is the time when a fiducial scale of interest first crosses the Hubble radius
during inflation, khc = a(thc )H(thc ). We find [22]
α=
2ω
H
4ω 2
β = −2 − ηss + ησσ −
,
3H 2
(7.35)
where , ησσ , and ηss are given by
Ḣ
H2
Mpl2 Mσσ
ησσ ≡
V
2
Mpl Mss
.
ηss ≡
V
≡−
(7.36)
The first two quantities function like the familiar slow-roll parameters from singlefield inflation: ησσ = 1 marks the end of the fields’ slow-roll evolution, after which
σ̈ ∼ H σ̇, while = 1 marks the end of inflation (ä = 0 for = 1). The third quantity,
ηss , is related to the effective mass of the isocurvature perturbations, and need not
remain small during inflation.
Using the transfer functions, we may relate the power spectra at thc to spectra at
later times. In the regime of interest, for late times and long wavelengths, we have
2
PR (k) = PR (khc ) 1 + TRS
(thc , t)
2
PS (k) = PR (khc ) TSS
(thc , t).
242
(7.37)
Ultimately, we may use TRS and TSS to calculate the isocurvature fraction,
βiso ≡
2
TSS
PS
= 2
,
2
PS + PR
TSS + TRS
+1
(7.38)
which may be compared to recent observables reported by the Planck collaboration.
An example of the fields’ trajectory of interest is shown in Fig. 7-3. As shown
in Fig. 7-4, while the fields evolve near the top of the ridge, the isocurvature modes
are tachyonic, µ2s < 0, leading to the rapid amplification of isocurvature modes.
When the turn-rate is nonzero, ω 6= 0, the growth of Qs can transfer power to
the adiabatic perturbations, Qσ . If TRS grows too large from this transfer, then
predictions for observable quantities such as ns can get pulled out of agreement with
present observations, as shown in the intermediate region of Fig. 7-2 and developed
in more detail in Section 7.4. On the other hand, growth of Qs is strongly suppressed
when fields evolve in a valley, since µ2s /H 2 1. In order to produce an appropriate
fraction of isocurvature perturbations while also keeping observables such as ns close
to their measured values, one therefore needs field trajectories that stay on a ridge for
a significant number of e-folds and have only a modest turn-rate so as not to transfer
too much power to the adiabatic modes. This may be accomplished in the regime of
weak curvature, κ 1.
7.3
Trajectories of Interest
7.3.1
Geometry of the Potential
As just noted, significant growth of isocurvature perturbations occurs when µ2s < 0,
when the fields begin near the top of a ridge. If the fields start in a valley, or if the
curvature near the top of the ridge is large enough (κ 1) so that the fields rapidly
fall into a valley, then the system quickly relaxes to the single-field attractor found in
[19], for which βiso → 0. To understand the implications for quantities such as βiso ,
it is therefore important to understand the geomtery of the potential. This may be
243
Figure 7-3: The fields’ trajectory (red) superimposed upon the effective potential in
the Einstein frame, V , with couplings ξφ = 1000, ξχ = 1000.015, λφ = λχ = g = 0.01,
and initial conditions φ0 = 0.35, χ0 = 8.1 × 10−4 , φ̇0 = χ̇0 = 0, in units of Mpl .
accomplished by working with the field-space coordinates r and θ, defined via
φ = r cos θ , χ = r sin θ.
(7.39)
(The parameter θ was labeled γ in [27].) Inflation in these models occurs for ξφ φ2 +
ξχ χ2 Mpl2 [22]. That limit corresponds to taking r → ∞, for which the potential
becomes
Vr→∞ (θ) =
Mpl4 2g cos2 θ sin2 θ + λφ cos4 θ + λχ sin4 θ
.
2
4
ξφ cos2 θ + ξχ sin2 θ
(7.40)
We further note that for our choice of potential in Eq. (7.7), V (φ, χ) has two discrete
symmetries, φ → −φ and χ → −χ. This means that we may restrict our attention
to only one quarter of the φ − χ plane. We choose φ > 0 and χ > 0 without loss of
generality.
The extrema (ridges and valleys) are those places where V,θ = 0, which formally
244
1.0
Μs 2 H2
HِHL‰103
0.8
0.6
0.4
0.2
0.0
-0.2
0
20
40
60
80
N*
Figure 7-4: The mass of the isocurvature modes, µ2s /H 2 (blue, solid), and the turn
rate, (ω/H) × 103 (red, dotted), versus e-folds from the end of inflation, N∗ , for the
trajectory shown in Fig. 7-3. Note that while the fields ride along the ridge, the
isocurvature modes are tachyonic, µ2s < 0, leading to an amplification of isocurvature
perturbations. The mass µ2s becomes large and positive once the fields roll off the
ridge, suppressing further growth of isocurvature modes.
has three solutions for 0 < θ < π/2 and r → ∞:
" p
#
Λχ
π
−1
p
θ1 = 0, θ2 = , θ3 = cos
,
2
Λφ + Λ χ
(7.41)
where we have defined the convenient combinations
Λφ ≡ λφ ξχ − gξφ
(7.42)
Λχ ≡ λχ ξφ − gξχ .
In order for θ3 to be a real angle (between 0 and π/2), the argument of the inverse
cosine in Eq. (7.41) must be real and bounded by 0 and 1. If Λχ and Λφ have the
same sign, both conditions are automatically satisfied. If Λχ and Λφ have different
signs then the argument may be either imaginary or larger than 1, in which case there
is no real solution θ3 . If both Λχ and Λφ have the same sign, the limiting cases are:
for Λχ Λφ , then θ3 → 0, and for Λχ Λφ then θ3 → π/2.
In each quarter of the φ−χ plane, we therefore have either two or three extrema, as
shown in Fig. 7-5. Because of the mean-value theorem, two ridges must be separated
245
V¥
0.5
1.0
1.5
2.0
2.5
3.0
Θ
Figure 7-5: The asymptotic value r → ∞ for three potentials with Λχ = −0.001
(blue dashed), Λχ = 0 (red solid), and Λχ = 0.001 (yellow dotted), as a function of
the angle θ = arctan(χ/φ). For all three cases, Λφ = 0.0015, ξφ = ξχ = 1000, and
λφ = 0.01.
by a valley and vice versa. If Λχ and Λφ have opposite signs, there are only two
extrema, one valley and one ridge. This was the case for the parameters studied in
[22]. If Λφ and Λχ have the same sign, then there is a third extremum (either two
ridges and one valley or two valleys and one ridge) within each quarter plane. In the
case of two ridges, their asymptotic heights are
Vr→∞ (θ1 ) =
λφ Mpl4
,
4ξφ2
λχ Mpl4
Vr→∞ (θ2 ) =
,
4ξχ2
(7.43)
and the valley lies along the direction θ3 . In the limit r → ∞, the curvature of the
potential at each of these extrema is given by
V,θθ |θ=0
V,θθ |θ=θ3
Λφ Mpl4
Λχ Mpl4
= − 3 , V,θθ |θ=π/2 = − 3 ,
ξφ
ξχ
2Λχ Λφ (Λφ + Λχ )2 Mpl4
=
.
(ξχ Λφ + ξχ Λχ )3
(7.44)
In this section we have ignored the curvature of the field-space manifold, since for
246
large field values the manifold is close to flat [22], and hence ordinary and covariant
derivatives nearly coincide. We demonstrate in Appendix B that the classification of
local curvature introduced here holds generally for the dynamics relevant to inflation,
even when one takes into account the nontrivial field-space manifold.
7.3.2
Linearized Dynamics
In this section we will examine trajectories for which ω is small but nonzero: small
enough so that the isocurvature perturbations do no transfer all their energy away
to the adiabatic modes, but large enough so that genuine multifield effects (such as
βiso 6= 0) persist rather than relaxing to effectively single-field evolution.
We focus on situations in which inflation begins near the top of a ridge of the
potential, with φ0 large and both χ0 and χ̇0 small. Trajectories for which the fields
remain near the top of the ridge for a substantial number of e-folds will produce
a significant amplification of isocurvature modes, since µ2s < 0 near the top of the
ridge and hence the isocurvature perturbations grow via tachyonic instability. From
a model-building perspective it is easy to motivate such initial conditions by postulating a waterfall transition, similar to hybrid inflation scenarios [38], that pins the
χ field exactly on the ridge. Anything from a small tilt of the potential to quantum
fluctuations would then nudge the field off-center.
With χ0 small, sufficient inflation requires ξφ φ20 Mpl2 , which is easily accomplished with sub-Planckian field values given ξφ 1. We set the scale for χ0 by
imagining that χ begins exactly on top of the ridge. In the regime of weak curvature,
κ 1, quantum fluctuations will be of order
where we take χrms ≡
H2
H
⇒ χrms = √
χ2 =
2π
2π
(7.45)
p
hχ2 i to be a classical estimator of the excursion of the field
away from the ridge. The constraint from Planck that H/Mpl ≤ 3.7 × 10−5 during
inflation then allows us to estimate χrms ∼ 10−5 Mpl at the start of inflation. (A
√
Gaussian wavepacket for χ will then spread as N , where N is the number of e-folds
247
of inflation.) This sets a reasonable scale for χ0 ; we examine the dynamics of the
system as we vary χ0 around χrms .
We may now expand the full background dynamics in the limit of small κ, χ, and
χ̇. The equation of motion for φ, given by Eq. (7.15), does not include any terms
linear in χ or χ̇, so the evolution of φ in this limit reduces to the single-field equation
of motion, which reduces to
φ̇SR
p
λφ Mpl3
'− √ 2
3 3ξφ φ
(7.46)
in the slow-roll limit [27]. To first approximation, the φ field rolls slowly along the
top of the ridge. Upon using Eq. (7.11), we may integrate Eq. (7.46) to yield
ξφ φ2∗
4
' N∗ ,
2
Mpl
3
(7.47)
where N∗ is the number of e-folds from the end of inflation, and we have used φ(t∗ ) φ(tend ). The slow-roll parameters may then be evaluated to lowest order in χ and χ̇
and take the form [19]
3
4N∗2
1
3
'−
1−
.
N∗
4N∗
'
ησσ
(7.48)
Expanding the equation of motion for the χ field and considering ξφ , ξχ 1 we
find the linearized equation of motion
χ̈ + 3H χ̇ −
Λφ Mpl2
χ ' 0,
ξφ2
"
s
(7.49)
which has the simple solution
χ(t) ' χ0 exp
3H
−
±
2
9H 2 Λφ Mpl2
+
2
ξφ2
where we again used Eq. (7.11) for H, and N (t) ≡
248
Rt
t0
!
#
N (t) ,
(7.50)
Hdt0 is the number of e-folds
since the start of inflation. If we assume that Λφ Mpl2 /ξφ2 9H 2 /4, which is equivalent
to Λφ /λφ 3/16, then we may Taylor expand the square root in the exponent of
χ(t). This is equivalent to dropping the χ̈ term from the equation of motion. In this
limit the solution becomes
χ(t) ' χ0 eκN (t) ,
(7.51)
where κ is defined in Eq. (7.10). Upon using the definition of Λφ in Eq. (7.42), we
now recognize κ = 4Λφ /λφ . Our approximation of neglecting χ̈ thus corresponds to
the limit κ 3/4.
When applying our set of approximations to the isocurvature mass in Eq. (7.29),
we find that the Mss term dominates ω 2 /H 2 , and the behavior of Mss in turn is
dominated by DJ DK V rather than the term involving RIJKL . Since we are projecting
the mass-squared matrix orthogonal to the fields’ motion, and since we are starting on
a ridge along the φ direction, the derivative of V that matters most to the dynamics
of the system in this limit is Dχχ V evaluated at small χ. To second order in χ, we
find
Dχχ V = −
Λφ Mpl4
ξφ3 φ2
Mpl6
(1 + 6ξφ )
+ 3
2Λφ
− λφ ε
ξφ (1 + 6ξφ )φ4
ξφ
h
Mpl4 χ2
+ 3
3(1 + 6ξφ )Λχ +
ξφ (1 + 6ξφ )φ4
(7.52)
+ (1 − ε)(1 + 6ξχ )Λφ
i
+ 6(1 − ε)(1 + 6ξφ )Λφ − Λφ ε ,
where we have used Λφ and Λχ as given in Eq. (7.42) and also introduced
ε≡
ξφ − ξχ
ξχ
=1− .
ξφ
ξφ
(7.53)
These terms each illuminate an aspect of the geometry of the potential: as we found
in Eq. (7.44), Λφ and Λχ are proportional to the curvature of the potential along the
249
φ and χ axes respectively, and ε is the ellipticity of the potential for large field values.
Intuition coming from these geometric quantities motivates us to use them as a basis
for determining the dynamics in our simulations. The approximations hold well for
the first several e-folds of inflation, before the fields fall off the ridge of the potential.
Based on our linearized approximation we may expand all kinematical quantities
in power series of χ0 and 1/N∗ . We refer to the intermediate quantities in Appendix B
and report here the important quantities that characterize the generation and transfer
of isocurvature perturbations. To lowest order in χ and χ̇, the parameter ηss defined
in Eq. (7.36) takes the form
3
ηss ' −κ −
4N∗
2ε
κ+
3
+
3
(1 − ε) ,
8N∗2
(7.54)
showing that to lowest order in 1/N∗ , ηss ∼ −κ < 0 and hence the isocurvature modes
begin with a tachyonic mass. The quantities α and β from Eq. (7.35) to first order
are
κ χ0 exp [κ(Ntot − N∗ )] p
√
N∗ ,
2 ξφ Mpl
1 3κ ε
1 3ε 9
β 'κ+
+ −1 + 2
−
,
N∗ 4
2
N∗ 8
8
α'
(7.55)
where Ntot is the total number of e-folds of inflation. These expansions allow us to
approximate the transfer function TSS of Eq. (7.34),
TSS '
N∗
Nhc
1− 3κ4 − 2ε
3
× exp κ (Nhc − N∗ ) − (3 − ε)
8
1
1
−
N∗ Nhc
(7.56)
,
where Nhc is the number of e-folds before the end of inflation at which Hubble crossing
occurs for the fiducial scale of interest. We may then use a semi-analytic form for
TRS by putting Eq. (7.56) into Eq. (7.34). This approximation is depicted in Fig.
7-6.
Our analytic approximation for TSS vanishes identically in the limit N∗ → 0 (at
250
4
Exact
Analytic
Exact
Semi-Analytic
0.035
3
0.030
TRS
TSS
0.025
2
1
0.020
0.015
0.010
0.005
0
0
10
20
30
N*
40
50
60
0.000
0
10
20
30
N*
40
50
60
Figure 7-6: The evolution of TSS (top) and TRS (bottom) using the exact and approximated expressions, for κ ≡ 4Λφ /λφ = 0.06, 4Λχ /λχ = −0.06 and ε = −1.5 × 10−5 ,
with φ0 = 0.35 Mpl , χ0 = 8.1 × 10−4 Mpl , and φ̇0 = χ̇0 = 0. We take Nhc = 60
and plot TSS and TRS against N∗ , the number of e-folds before the end of inflation.
The approximation works particularly well at early times and matches the qualitative
behavior of the exact numerical solution at late times.
the end of inflation), though it gives an excellent indication of the general shape of
TSS for the duration of inflation. We further note that TSS is independent of χ0 to
lowest order, while TRS ∝ α ∝ κχ0 and hence remains small in the limit we are
considering. Thus for small κ, we expect βiso to be fairly insensitive to changes in χ0 .
7.4
Results
We want to examine how the isocurvature fraction βiso varies as we change the shape of
the potential. We are particularly interested in the dependence of βiso on κ, since the
leading-order contribution to the isocurvature fraction from the shape of the potential
is proportional to κ. Guided by our approximations, we simulated trajectories across
1400 potentials and we show the results in Figures 7-7 - 7-10. The simulations were
done using zero initial velocities for φ and χ, and were performed using both Matlab
and Mathematica, as a consistency check. We compare analytical approximations in
certain regimes with our numerical findings.
As expected, we find that there is an interesting competition between the degree
to which the isocurvature mass is tachyonic and the propensity of the fields to fall off
the ridge. More explicitly, for small κ we expect the fields to stay on the ridge for
most of inflation with a small turn rate that transfers little power to the adiabatic
251
modes. Therefore, in the small-κ limit, TRS remains small while TSS (and hence
βiso ) increases exponentially with increasing κ. Indeed, all the numerical simulations
show that βiso vs. κ increases linearly on a semilog scale for small κ. However, in
the small-κ limit, the tachyonic isocurvature mass is also small, so βiso remains fairly
small in that regime. Meanwhile, for large κ we expect the fields to have a larger
tachyonic mass while near the top of the ridge, but to roll off the ridge (and transfer
significant power to the adiabatic modes) earlier in the evolution of the system. There
should be an intermediate regime of κ in which the isocurvature mass is fairly large
(and tachyonic) and the fields do not fall off the ridge too early. Indeed, a ubiquitous
feature of our numerical simulations is that βiso is always maximized around κ . 0.1,
regardless of the other parameters of the potential.
7.4.1
Local curvature of the potential
In Fig. 7-7, we examine the variation of βiso as we change χ0 and κ. As expected,
βiso has no dependence on χ0 for small κ. Increasing κ breaks the χ0 degeneracy: the
closer the fields start to the top of the ridge, the more time the fields remain near the
top before rolling off the ridge and transferring power to the adiabatic modes. Just
as expected, for the smallest value of χ0 , we see the largest isocurvature fraction.
Even for relatively large χ0 , there is still a nontrivial contribution of isocurvature
modes to the perturbation spectrum. Therefore, our model generically yields a large
isocurvature fraction with little fine-tuning of the initial field values in the regime
κ 1.
We may calculate βiso for the limiting case of zero curvature, κ → 0, the vicinity
in which the curves in Fig. 7-7 become degenerate. Taking the limit κ → 0 means
essentially reverting to a Higgs-like case, a fully SO(2) symmetric potential with no
turning of the trajectory in field space [27]. As expected, our approximate expression
in Eq. (7.56) for TRS → 0 in the limit κ → 0, and hence we need only consider TSS .
As noted above, our approximate expression for TSS in Eq. (7.56) vanishes in
the limit N∗ → 0. Eq. (7.56) was derived for the regime in which our approximate
expressions for the slow-roll parameters and ησσ in Eq. (7.48) are reasonably ac252
1
Χ0 = 10-4
Χ0 = 10-3
0.1
Χ0 = 10-2
Βiso
0.01
0.001
10-4
10-5
10-6
0.00
0.05
0.10
Κ
0.15
0.20
Figure 7-7: The isocurvature fraction for different values of χ0 (in units of Mpl ) as a
function of the curvature of the ridge, κ. All of the trajectories began at φ0 = 0.3 Mpl ,
which yields Ntot = 65.7. For these potentials, ξφ = 1000, λφ = 0.01, ε = 0, and
Λχ = 0. The trajectories that begin closest to the top of the ridge have the largest
values of βiso , with some regions of parameter space nearly saturating βiso = 1.
curate. Clearly the expressions in Eq. (7.48) will cease to be accurate near the end
of inflation. Indeed, taking the expressions at face value, we would expect slow roll
√
to end (|ησσ | = 1) at N∗ = 1/2, and inflation to end ( = 1) at N∗ = 2/ 3, rather
than at N∗ = 0. Thus we might expect Eq. (7.48) to be reliable until around N∗ ' 1,
which matches the behavior we found in a previous numerical study [19]. Hence we
will evaluate our analytic approximation for TSS in Eq. (7.56) between Nhc = 60 and
N∗ ' 1, rather than all the way to N∗ → 0. In the limit κ → 0 and ε → 0 and using
N∗ = 1, Eq. (7.56) yields
TSS '
1
exp [−9/8] ,
Nhc
(7.57)
upon taking Nhc N∗ . For Nhc = 60, we therefore find TSS ' 5.4 × 10−3 , and
hence βiso ' 2.9 × 10−5 . This value may be compared with the exact numerical value,
βiso = 2.3 × 10−5 . Despite the severity of our approximations, our analytic expression
provides an excellent guide to the behavior of the system in the limit of small κ.
As we increase κ, the fields roll off the ridge correspondingly earlier in their evolution. The nonzero turn-rate causes a significant transfer of power from the isocurvature modes to the adiabatic modes. As TRS grows larger, it lowers the overall value
253
10
Β iso
TSS 2
TRS 2
0.1
Βiso
0.001
10-5
10-7
10-9
0.00
0.05
0.10
Κ
0.15
0.20
Figure 7-8: Contributions of TRS and TSS to βiso . The parameters used are φ0 =
0.3 Mpl , χ0 = 10−3 Mpl , ξφ = 103 , λφ = 0.01, ε = 0 and Λχ = 0. For small κ, βiso is
dominated by TSS ; for larger κ, TRS becomes more important and ultimately reduces
βiso .
of βiso . See Fig. 7-8.
7.4.2
Global structure of the potential
The previous discussion considered the behavior for Λχ = 0. As shown in Fig. 7-5,
the global structure of the potential will change if Λχ 6= 0. In the limit κ 1, the
fields never roll far from the top of the ridge along the χ = 0 direction, and therefore
the shape of the potential along the χ direction has no bearing on βiso . However,
large κ breaks the degeneracy in Λχ because the fields will roll off the original ridge
and probe features of the potential along the χ direction. See Fig. 7-9.
In the case Λχ = 0, the fields roll off the ridge and eventually land on a plain,
where the isocurvature perturbations are minimally suppressed, since µ2s ∼ 0. For
Λχ > 0, there is a ridge along the χ direction as well as along χ = 0, which means
that there must be a valley at some intermediate angle in field space. When the
fields roll off the original ridge, they reach the valley in which µ2s > 0, and hence the
isocurvature modes are more strongly suppressed than in the Λχ = 0 case.
Interesting behavior may occur for the case Λχ < 0. There exists a range of κ
for which the isocurvature perturbations are more strongly amplified than a naive
254
0.1
L Χ = -0.001
LΧ = 0
L Χ = 0.001
Βiso
10-4
10-7
10-10
10-13
0.00
0.05
0.10
Κ
0.15
0.20
Figure 7-9: The isocurvature fraction for different values of Λχ as a function of the
curvature of the ridge, κ. All of the trajectories began at φ0 = 0.3 Mpl and χ0 =
10−4 Mpl , yielding Ntot = 65.7. For these potentials, ξφ = 1000, λφ = 0.01, and
ε = 0. Potentials with Λχ < 0 yield the largest βiso peaks, though in those cases
βiso falls fastest in the large-κ limit due to sensitive changes in curvature along the
trajectory. Meanwhile, potentials with positive Λχ suppress the maximum value of
βiso once κ & 0.1 and local curvature becomes important.
estimate would suggest, thanks to the late-time behavior of ηss ∼ (Dχχ V )/V . If the
second derivative decreases more slowly than the potential itself, then the isocurvature
modes may be amplified for a short time as the fields roll down the ridge. This added
contribution is sufficient to increase βiso compared to the cases in which Λχ ≥ 0.
However, the effect becomes subdominant as the curvature of the original ridge, κ,
is increased. For larger κ, the fields spend more time in the valley, in which the
isocurvature modes are strongly suppressed.
In Figure 7-10, we isolate effects of ε and κ on βiso . From Eq. (7.52), when Λφ is
small (which implies that κ is small), ε sets the scale of the isocurvature mass. Positive
ε makes the isocurvature mass-squared more negative near κ = 0, which increases the
power in isocurvature modes. Conversely, negative ε makes the isocurvature masssquared less negative near κ = 0, which decreases the power in isocurvature modes.
In geometrical terms, in the limit Λφ = Λχ = 0, equipotential surfaces are ellipses
p
√
with eccenticity ε for ε > 0 and ε/(ε − 1) for ε < 0. In this limit we may calculate
βiso exactly as we did for the case of ε = 0.
The other effect of changing ε is that it elongates the potential in either the φ
255
or χ direction. This deformation of the potential either enhances or decreases the
degree to which the fields can turn, which in turn will affect the large-κ behavior. In
particular, for ε > 0 the potential is elongated along the φ direction, which means
that when the fields roll off the ridge, they immediately start turning and transferring
power to the adiabatic modes. Conversely, for ε < 0 the potential is elongated along
the χ direction, so once the fields fall off the ridge, they travel farther before they
start turning. Therefore, in the large-κ limit, βiso falls off more quickly for ε > 0 than
for ε < 0.
¶ = 12
¶=0
¶ = -12
0.1
Βiso
0.001
10-5
10-7
0.00
0.05
0.10
Κ
0.15
0.20
Figure 7-10: The isocurvature fraction for different values of ε as a function of the
curvature of the ridge, κ. All of the trajectories began at χ0 = 10−3 Mpl and φ0 =
0.3 Mpl , with Ntot = 65.7. For these potentials, ξφ = 1000, λφ = 0.01, and Λχ = 0.
Here we see the competition between ε setting the scale of the isocurvature mass and
affecting the amount of turning in field-space.
We may use our analytic expression for TSS in Eq. (7.56) for the case in which
2
κ → 0 with ε 6= 0. We find the value of βiso ' TSS
changes by a factor of 11 when
we vary ε ± 1/2, while our numerical solutions in Fig. 7-10 vary by a factor of 21.
Given the severity of some of our analytic approximations, this close match again
seems reassuring.
7.4.3
Initial Conditions
The quantity βiso varies with the fields’ initial conditions as well as with the parameters of the potential. Given the form of TRS and TSS in Eq. (7.34), we see that the
256
value of βiso depends only on the behavior of the fields between Nhc and the end of
inflation. This means that if we were to change φ0 and χ0 in such a way that the
fields followed the same trajectory following Nhc , the resulting values for βiso would
be identical.
We have seen in Eq. (7.47) that we may use φ as our inflationary clock, ξφ φ2∗ /Mpl2 '
4N∗ /3, where N∗ = Ntot − N (t) is the number of e-folds before the end of inflation.
We have also seen, in Eq. (7.51), that for small κ we may approximate χ(t) '
χ0 exp[κN (t)]. If we impose that two such trajectories cross Nhc with the same value
of χ, then we find
3
∆(log χ0 ) = κ∆N = − ξφ κ ∆
4
φ20
Mpl2
!
.
(7.58)
We tested the approximation in Eq. (7.58) by numerically simulating over 15,000
trajectories in the same potential with different initial conditions. The numerical
results are shown in Fig. 7-11, along with our analytic predictions, from Eq. (7.58),
that contours of constant βiso should appear parabolic in the semilog graph. As
shown in Fig. 7-11, our analytic approximation matches the full numerical results
remarkably well. We also note from Fig. 7-11 that for a given value of χ0 , if we
increase φ0 (thereby increasing the total duration of inflation, Ntot ), we will decrease
βiso , behavior that is consistent with our approximate expressions for TRS and TSS in
Eq. (7.56).
7.4.4
CMB observables
Recent analyses of the Planck data for low multipoles suggests an improvement of fit
between data and underlying model if one includes a substantial fraction of primordial isocurvature modes, βiso ∼ O(0.1). The best fits are obtained for isocurvature
perturbations with a slightly blue spectral tilt, nI ≡ 1 + d ln PS /d ln k ≥ 1.0 [11]. In
the previous sections we have demonstrated that our general class of models readily produces βiso ∼ O(0.1) in the regime κ . 0.1. The spectral tilt, nI , for these
257
Log10HΒisoL
-4.0
-0.5
Log10H Χ0L
-4.5
-0.75
-1
-5.0
-2
-5.5
-3
-6.0
0.30
0.32
0.34
Φ0
0.36
0.38
-6
-4
-2
0246
Figure 7-11: Numerical simulations of βiso for various initial conditions (in units of
Mpl ). All trajectories shown here were for a potential with κ = 4Λφ /λφ = 0.116,
4Λχ /λχ = −160.12, and ε = −2.9 × 10−5 . Also shown are our analytic predictions
for contours of constant βiso , derived from Eq. (7.58) and represented by dark, solid
lines. From top right to bottom left, the contours have βiso = 0.071, 0.307, 0.054,
0.183, and 0.355.
perturbations goes as [14, 18]
nI = 1 − 2 + 2ηss ,
(7.59)
where and ηss are evaluated at Hubble-crossing, Nhc . Given our expressions in Eqs.
(7.48) and (7.54), we then find
3
nI ' 1 − 2κ −
2N∗
2ε
3
(1 + ε) .
κ+
−
3
4N∗2
(7.60)
For trajectories that produce a nonzero fraction of isocurvature modes, the isocurvature perturbations are tachyonic at the time of Hubble-crossing, with ηss ∝ Mss ∼
µ2s < 0. Hence in general we find nI will be slightly red-tilted, nI ≤ 1. However, in
the regime of weak curvature, κ 1, we may find nI ∼ 1. In particular, in the limit
κ → 0 and ε → 0, then nI → 1 − 3/(4N∗2 ) ∼ 1 − O(10−4 ), effectively indistinguishable
258
Figure 7-12: Two trajectories from Fig. 7-11 that lie along the βiso = 0.183 line, for
φ0 = 0.3 Mpl and φ0 = 0.365 Mpl . The dots mark the fields’ initial values. The two
trajectories eventually become indistinguishable, and hence produce identical values
of βiso .
from a flat, scale-invariant spectrum. In general for κ < 0.02, we therefore expect
nI > ns , where ns ∼ 0.96 is the spectral index for adiabatic perturbations. In that
regime, the isocurvature perturbations would have a bluer spectrum than the adiabatic modes, albeit not a genuinely blue spectrum. An important test of our models
will therefore be if future observations and analysis require nI > 1 in order to address the present low-` anomaly in the Planck measurements of the CMB temperature
anisotropies.
Beyond βiso and nI , there are other important quantities that we need to address,
and that can be used to distinguish between similar models: the spectral index for
the adiabatic modes, ns , and its running, α ≡ dns /d ln k; the tensor-to-scalar ratio, r;
and the amplitude of primordial non-Gaussianity, fNL . As shown in [19], in the limit
of large curvature, κ 1, the system quickly relaxes to the single-field attractor for
which 0.960 ≤ ns ≤ 0.967, α ∼ O(10−4 ), 0.0033 ≤ r ≤ 0.0048, and |fNL | 1. (The
ranges for ns and r come from considering Nhc = 50 − 60.) Because the single-field
259
attractor evolution occurs when the fields rapidly roll off a ridge and remain in a
valley, in which µ2s > 0, the models generically predict βiso 1 in the limit κ 1 as
well. Here we examine how these observables evolve in the limit of weak curvature,
κ 1, for which, as we have seen, the models may produce substantial βiso ∼ O(0.1).
Let us start with the spectral index, ns . If isocurvature modes grow and transfer
substantial power to the adiabatic modes before the end of inflation, then they may
affect the value of ns . In particular, we have [14, 18, 22]
ns = ns (thc ) +
1
[−α(thc ) − β(thc )TRS ] sin(2∆),
H
(7.61)
where
ns (thc ) = 1 − 6 + 2ησσ
(7.62)
and α and β are given in Eq. (7.35). The angle ∆ is defined via
TRS
cos ∆ ≡ p
.
2
1 + TRS
(7.63)
The turn rate α = 2ω/H is small at the moment when perturbations exit the Hubble
radius, and the trigonometric factor obeys −1 ≤ sin(2∆) ≤ 1. We also have β '
κ+O(N∗−1 ) at early times, from Eq. (7.55). Hence we see that TRS must be significant
in order to cause a substantial change in ns compared to the value at Hubble crossing,
ns (thc ). Yet we found in Fig. 7-8 that TRS grows large after βiso has reached its
maximum value. We therefore expect ns to be equal to its value in the single-field
attractor for κ . 0.1.
This is indeed what we find when we study the exact numerical evolution of ns
over a wide range of κ, as in Fig. 7-2, as well as in the regime of weak curvature,
κ 1, as shown in Fig. 7-13. For κ . 0.1 and using Nhc = 60, we find ns well
within the present bounds from the Planck measurements: ns = 0.9603 ± 0.0073
[11]. Moreover, because the regime κ . 0.1 corresponds to TRS 1, the analysis
of the running of the spectral index, α, remains unchanged from [19], and we again
find α ∼ −2/N∗2 ∼ O(10−4 ), easily consistent with the constraints from Planck,
260
0.98
ns
0.97
0.96
Planck 1Σ
0.95
Planck 2Σ
0.94
0.93
0.92
0.00
0.05
0.10
Κ
0.15
0.20
Figure 7-13: The spectral index ns for different values of the local curvature κ. The
parameters used are φ0 = 0.3 Mpl , χ0 = 10−3 Mpl , ξφ = 1000, λφ = 0.01, ε = 0 and
Λχ = 0. Comparing this with Fig. 7-7 we see that the peak in the βiso curve occurs
within the Planck allowed region.
α = −0.0134 ± 0.0090 [11].
Another important observational tool for distinguishing between inflation models
is the value of the tensor-to-scalar ratio, r. Although the current constraints are at
the 10−1 level, future experiments may be able to lower the sensitivity by one or two
orders of magnitude, making exact predictions potentially testable. For our models
the value of r is given by [19]
r=
16
.
2
1 + TRS
(7.64)
We see that once TRS ∼ O(1), the value of r decreases, as is depicted in Fig. 7-14.
One possible means to break the degeneracy between this family of models, apart
from βiso , is the correlation between r and ns . In the limit of vanishing TRS , both ns
and r revert to their single-field values, though they both vary in calculable ways as
TRS grows to be O(1). See Fig. 7-15.
We studied the behavior of fNL in our family of models in detail in [22]. There we
found that substantial fNL required a large value of TRS by the end of inflation. In
this chapter we have found that TRS remains small in the regime of weak curvature,
κ . 0.1. Using the methods described in detail in [22], we have evaluated fNL
numerically for the broad class of potentials and trajectories described above, in the
261
r
0.002
0.001
5 ´ 10-4
2 ´ 10-4
0.00
0.05
0.10
Κ
0.15
0.20
Figure 7-14: The tensor-to-scalar ratio as a function of the local curvature parameter
κ. The parameters used are φ0 = 0.3 Mpl , χ0 = 10−3 Mpl , ξφ = 1000, λφ = 0.01, ε = 0
and Λχ = 0.
0.00305
0.00300
r
0.00295
0.00290
0.00285
0.00280
0.00275
0.955
0.960
ns
0.965
Figure 7-15: The correlation between r and ns could theoretically break the degeneracy between our models. The parameters used for this plot are φ0 = 0.3 Mpl ,
χ0 = 10−3 Mpl , ξφ = 1000, λφ = 0.01, ε = 0 and Λχ = 0, with 0 ≤ κ ≤ 0.1.
262
limit of weak curvature (κ 1), and we find |fNL | O(1) for the entire range of
parameters and initial conditions, fully consistent with the latest bounds from Planck
[35].
Thus we have found that there exists a range of parameter space in which multifield dynamics remain nontrivial, producing βiso ∼ O(0.1), even as the other important observable quantities remain well within the most-favored region of the latest
observations from Planck.
7.5
Conclusions
Previous work has demonstrated that multifield inflation with nonminimal couplings
provides close agreement with a number of spectral observables measured by the
Planck collaboration [19] (see also [40]). In the limit of strong curvature of the effective
potential in the Einstein frame, κ 1, the single-field attractor for this class of models
pins the predicted value of the spectral index, ns , to within 1σ of the present best-fit
observational value, while also keeping the tensor-to-scalar ratio, r, well below the
present upper bounds. In the limit of κ 1, these models also generically predict no
observable running of the spectral index, and (in the absence of severe fine-tuning of
initial conditions [22]) no observable non-Gaussianity, |α|, |fNL | 1. In the limit of
the single-field attractor, however, these models also predict no observable multifield
effects, such as amplification of primordial isocurvature modes, hence βiso ∼ 0 in the
limit κ 1.
In this chapter, we have demonstrated that the same class of models can produce
significant isocurvature modes, βiso ∼ O(0.1), in the limit of weak curvature of the
Einstein-frame potential, κ ≤ 0.1. In that limit, these models again predict values for
ns , α, r, and fNL squarely within the present best-fit bounds, while also providing a
plausible explanation for the observed anomaly at low multipoles in recent measurements of CMB temperature anisotropies [11]. These models predict non-negligible
isocurvature fractions across a wide range of initial field values, with a dependence
of βiso on couplings that admits an analytic, intuitive, geometric interpretation. Our
263
geometric approach provides an analytically tractable method in excellent agreement
with numerical simulations, which could be applied to other multifield models in
which the effective potential is “lumpy.”
The mechanism for generating βiso ∼ 0.1 that we have investigated in this chapter
is based on the idea that a symmetry among the fields’ bare couplings λI , g, and ξI is
softly broken. Such soft breaking would result from a coupling of one of the fields (say,
χ) to either a CDM scalar field or to a neutrino species; some such coupling would
be required in order for the primordial isocurvature perturbations to survive to the
era of photon decoupling, so that the primordial perturbations could be impressed in
the CMB [20]. Hence whatever couplings might have enabled primordial isocurvature
modes to modify the usual predictions from the simple, purely adiabatic ΛCDM model
might also have generated weak but nonzero curvature in the effective potential,
κ 1. If the couplings λI , g, and ξI were not subject to a (softly broken) symmetry,
or if the fields’ initial conditions were not such that the fields began near the top of
a ridge in the potential, then the predictions from this class of models would revert
to the single-field attractor results analyzed in detail in [19].
Inflation in this class of models ends with the fields oscillating around the global
minimum of the potential. Preheating in such models offers additional interesting
phenomena [37], and further analysis is required to understand how the primordial
perturbations analyzed here might be affected by preheating dynamics. In particular,
preheating in multifield models — under certain conditions — can amplify perturbations on cosmologically interesting length scales [43]. Thus the behavior of isocurvature modes during preheating [44] requires careful study, to confirm whether preheating effects in the family of models considered here could affect any of the predictions
for observable quantities calculated in this chapter. We are presently studying effects
of preheating in this family of models.
Finally, expected improvements in observable constraints on the tensor-to-scalar
ratio, as well as additional data on the low-` portion of the CMB power spectrum,
could further test this general class of models and perhaps distinguish among members
of the class.
264
7.6
Acknowledgments
It is a pleasure to thank Bruce Bassett, Rhys Borchert, Xingang Chen, Alan Guth,
Carter Huffman, Scott Hughes, Adrian Liu, and Edward Mazenc for helpful discussions. This work was supported in part by the U.S. Department of Energy (DOE)
under Contract DE-FG02-05ER41360 and in part by MIT’s Undergraduate Research
Opportunities Program.
7.7
7.7.1
Appendix
Approximated Dynamical Quantities
In this appendix, we present results for dynamical quantities under our approximations that ξφ , ξχ 1, ξφ φ2 Mpl2 , and χ0 Mpl .
First we expand quantities associated with field-space curvature, starting with the
field-space metric, GIJ , using the definition from Eq. (7.6). We arrive at the following
expressions:
Gφφ
6Mpl2
'
φ2
Gφχ = Gχφ
Gχχ '
We also find
G φφ '
6Mpl2 ξχ χ
'
ξφ φ3
Mpl2
.
ξφ φ2
φ2
6Mpl2
G φχ = G χφ ' −
G χχ '
(7.65)
ξφ φ2
.
Mpl2
265
ξχ φχ
Mpl2
(7.66)
Next we expand the field-space Christoffel symbols, ΓIJK , and find
Γφφφ ' −
1
φ
Γφχφ = Γφφχ ' −
ξχ χ
ξφ φ2
ξχ
ξφ φ
ξχ χ
'
ξφ φ2
Γφχχ '
Γχφφ
(7.67)
1
φ
ξχ χ (2ξφ − ξχ )
'−
.
ξφ2 φ2
Γχχφ = Γχφχ ' −
Γχχχ
The nonzero components of the field-space Riemann curvature tensor become
Rφφφχ = −Rφφχφ ' ε(ε − 1)
χ
φ3
ε
6ξφ φ2
ε
'− 2
φ
Rφχφχ = −Rφχχφ ' −
Rχφχφ
=
−Rχφφχ
Rχχφχ = −Rχχχφ ' ε(1 − ε)
(7.68)
χ
.
φ3
We also expand dynamical quantities, beginning with the fields’ velocity:
p
σ̇ '
2λφ Mpl4
,
3ξφ2 φ2
(7.69)
and the turn rate ω in the φ and χ directions:
ωφ ' 0
p
2
3φ
2M
Λ
χ
−
3λ
ξ
χ̇
pl
φ
φ
φ
p
ωχ '
.
2 2λφ Mpl3
266
(7.70)
7.7.2
Covariant formalism and potential topography
We have defined the character of the maxima and minima of the potential using the
(normal) partial derivative at asymptotically large field values, where the manifold
is asymptotically flat, hence the normal and covariant derivatives asymptote to the
same value. By keeping the next to leading order term in the series expansion, we
can test the validity of this approach for characterizing the nature of the extrema.
We take as an example the potential parameters used in Fig. 7-3, specifically
ξφ = 1000, ξχ = 999.985, λφ = 0.01, λχ = 0.01, g = 0.01. The ridge of the potential
occurs at χ = 0.
The asymptotic value of the second partial derivative is
V,χχ |χ=0 →
−Mpl4 Λφ
−Mpl4 × 1.5 · 10−5
=
ξφ3 φ2
ξφ φ2
(7.71)
Let us look at the partial second derivative for χ = 0 and finite φ:
V,χχ |χ=0
2
2
−Λ
ξ
φ
+
gξ
M
φ
φ
φ
pl
= Mpl4 φ2
ξφ (Mpl2 + ξφ φ2 )3
∝ −0.015 ξφ φ2 + 10Mpl2 .
(7.72)
We see that the two terms can be comparable. In particular, the second derivative
changes sign at
V,χχ |χ=0 = 0 ⇒ ξφ φ2tr ≈ 667Mpl2
(7.73)
which is a field value larger than the one we used for our calculation. In order to get
70 efolds of inflation, ξφ φ2 ∼ 100Mpl2 , significantly smaller than the transition value.
For φ < φtr the second derivative is positive, meaning there is a transition where the
local maximum becomes a local minimum. This means that if one was to take our
Einstein frame potential as a phenomenological model without considering the field
space metric, even at large field values, where slow roll inflation occurs, the results
would be qualitatively different.
Let us now focus our attention on the covariant derivative, keeping in mind that
267
in a curved manifold it is a much more accurate indicator of the underlying dynamics.
Dχχ V = V,χχ − Γφχχ V,φ − Γχχχ V,χ .
(7.74)
Looking at the extra terms and keeping the lowest order terms we have V,χ = 0 by
symmetry, V,φ ≈ λφ /(ξφ3 φ3 ), and Γφχχ = ξφ (1 + 6ξχ )φ/C ≈ ξχ /(ξφ φ).
We will now expand the covariant derivative term in 1/φ and also in ξφ and ξχ .
This way we will make sure that there is no transition in the behavior of the extremum
for varying field values, that is to say the character of the extremum will be conserved
term by term in the expansion (we only show this for the first couple of terms, but
the trend is evident). We find
Dχχ V =
−Λφ Mpl4
ξφ3 φ2
Mpl6
λφ ε
1
2Λφ −
+ 3 2
1−
+ ...
ξφ φ (ξφ φ2 )
6
6ξφ
+
Mpl8
×
ξφ3 φ2 (ξφ φ2 )2
λφ
λφ
−3Λφ + (1 + 2ε) −
(1 + ε) + ...
6
36ξφ
(7.75)
+ ...
We have written the covariant derivative using the geometrically intuitive combinations of parameters, which was done in the main text in a more general setting
(χ 6= 0). It is worthwhile to note that we did not write the closed form solution
for Dχχ V (which is straightforward to calculate using the Christoffel symbols, given
explicitly in [22]), since this power series expansion is both more useful and more
geometrically transparent, since it is easy to see the order at which each effect is first
introduced.
We see that once we take out the (1/ξφ3 φ2 ) behavior there remains a multiple series
expansion as follows
• Series in (1/ξφ φ2 )
268
• Each term of the above series is expanded in inverse powers of ξφ .
For the example of Fig. 7-3 the relevant quantity that defines to lowest order in
ξφ and ξχ all terms of the series is Λφ = 0.015.
By inspection of the terms, we can see that for our choice of parameters the first
term defines the behavior of the covariant derivative, which is also the asymptotic
value of the normal second derivative that we used to characterize the character of
the extremum. In the case when Λφ = 0 the ellipticity term e is dominant. Even if
λφ = ε = 0 then the dominant term comes at an even higher order and is proportional
to λφ .
In other words, the character of the extremum is conserved if one considers the
covariant derivatives. For asymptotically large field values the two coincide, since the
curvature vanishes. It is thus not only quantitatively but also qualitatively essential
to use our covariant formalism for the study of these models, even at large field values
where the curvature of the manifold is small.
Now that the character of the maximum is clear we can proceed to calculating all
ηss . We neglect the term in Mss that is proportional to RIJKL , since the curvature
of the field-space manifold is subdominant for ξφ φ20 Mpl2 and the RIJKL term is
multiplied by two factors of the fields’ velocity. If in addition we take χ = χ̇ = 0,
then Mss becomes
ξφ φ2
Mss ' ŝ ŝ Dχχ V =
Mpl2
χ χ
Mpl2
1+
Dχχ V.
ξφ φ2
(7.76)
Using the double series expansion of Eq. (7.75) the entropic mass-squared becomes
Mss V =
−Λφ Mpl2
ξφ2
Mpl4
λφ ε
1
Λφ −
+ 2
1−
+ ...
ξφ (ξφ φ2 )
6
6ξφ
Mpl6
λφ
+ 2
−Λφ + (1 + ε) + ... + ...
ξφ (ξφ φ2 )2
6
(7.77)
To find the generalized slow roll parameter ηss we need to divide by the potential,
269
which again can be expanded in a power series for χ → 0 as
V = Mpl4
λφ
3λφ
λφ
− Mpl6 3 2 + Mpl8 4 4 + ...
2
4ξφ
2ξφ φ
4ξφ φ
(7.78)
The calculation of ηss is now a straightforward exercise giving
Mpl2 Mss
−4Λφ
=
V
λφ
2 Mpl −4Λφ 2ε
1
+
−
+O
2
ξφ φ
λφ
3
ξφ
4
Mpl
2
1
(1
−
ε)
+
O
+
(ξφ φ2 )2 3
ξφ
1
+O
(ξφ φ2 )3
3
2ε
9
2
(1 − ε)
≈ −κ +
−κ −
+
4N∗
3
16N∗2 3
ηss ≈
(7.79)
where we used the slow-roll solution for φ from Eq. (7.47), identifying it as the
inflationary clock and the definition κ = 4Λφ /λφ . By setting κ = ε = 0 we see that
even in the fully symmetric case the isocurvature mass is small but positive.
In the limit of χ → 0 there is no turning (ω = 0), and hence TRS = 0. In order to
calculate TSS we need
β = −2 − ηss + ησσ
1 3κ ε
1 3ε 9
' κ+
+ −1 + 2
−
.
N∗ 4
2
N∗ 8
8
(7.80)
From Eq. (7.34), we see that TSS depends on the integral
Z
t
0
Z
Nhc
βHdt =
thc
N∗
270
βdN 0 .
(7.81)
Plugging in the expression for β from Eq. (7.80)
Z
Nhc
N∗
βdN 0 = κ(Nhc − N∗ )
Nhc
1
1
− c1 ln
− c2
−
N∗
N∗ Nhc
(7.82)
where
3κ ε
−
4
2
9 3ε
=
− .
8
8
c1 = 1 −
(7.83)
c2
(7.84)
Of course there is the ambiguity of stopping the integration one e-fold before the end
of inflation. If one plots β vs. N∗ and does a rough integration of the volume under
R0
the curve, one finds this area giving an extra contribution 1 βdN ∼ −1. This is a
change, but not a severe one. We will neglect it for now, keeping in mind that there is
an O(1) multiplicative factor missing from the correct result. However since β varies
over a few orders of magnitude, we can consider this factor a small price to pay for
such a simple analytical result.
271
272
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